CONTENTS COLLEGE ALGEBRA: DR.YOU

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1 1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p. LECTURE 1-5. SOLVING OTHER EQUATIONS A.6 p.41 LECTURE 1-7 DISTANCE, MIDPOINT, CIRCLES 1.1, 1.4 p.49 REVIEW PROBLEMS p.56 UNIT 1 MORE PRACTICE PROBLEMS UNIT LECTURE -1 INTERCEPTS, SYMMETRY, FUNCTIONS 1.,.1 p.60 LECTURE - PROPERTIES OF FUNCTIONS. p.69 LECTURE - MORE ABOUT FUNCTIONS.1,.4 p.77 LECTURE -4 OPERATIONS OF FUCTIONS.1 p.85 LECTURE -5 TRANSFORMATIONS.5 p.9 REVIEW PROBLEMS p.100 UNIT MORE PRACTICE PROBLEMS UNIT LECTURE -1 LINES 1.4 p.105 LECTURE - RELATION OF TWO LINES.1 p.115 LECTURE - SYSTEM OF LIEAR EQUATIONS 11.1 p.1 LECTURE -4 QUADRATIC FUNCTIONS. p.14 REVIEW PROBLEMS p.150 UNIT MORE PRACTICE PROBLEMS UNIT 4 LECTURE 4-1 POLYNOMIALS 4.1 p.156 LECTURE 4- DIVISION OF POLYNOMIALS A. p.16 LECTURE 4- ZEROS OF POLYNOMIALS 4., 4. p.17 LECTURE 4.4 RATIONAL FUNCTION 4.4, 4.5 p.18 LECTURE 4-5 INEQUALITIES.5, 4.6 p.19 REVIEW PROBLEMS p.01 UNIT 4 MORE PRACTICE PROBLEMS UNIT 5 LECTURE 5-1 COMPOSITION OF FUNCTIONS 5.1 p.05 LECTURE 5- INVERSE/ONE TO ONE FUNCTIONS 5. p.1 LECTURE 5- EXPONENTIAL/LOG FUNCTIONS 5., 5.4 p. LECTURE 5-4 EXPONENTIAL EQUATIONS 5., 5.6 p.5 LECTURE 5-5 LOGARITHMIC EQUATIONS 5.6 p.45 LECTURE 5-6. APPLICATIONS 5.5 p.5 REVIEW PROBLEMS 5.7, 5.8 p.57 UNIT 5 MORE PRACTICE PROBLEMS FINAL PRACTICE FOR FINAL p.6

2 UNIT 1. ALGEBRAIC EXPRESSIONS AND CIRCLES LECTURE 1-1 REVIEW Video 1) Simple exponent properties ) Negative exponent properties ) Simplifying algebraic expressions BASE AND EXPONENT x = x x x = x x x factors of x x base exponent (this is read x to the -th power ) NOTATION: R means all real numbers PROPERTIES OF EXPONENTS: Let m, n R 1) a n a m = a n+m Product rule ) (a n ) m = a n m Power rule ) (ab) n = a n b n Power of product 4) a n a m = an am = an m Quotient 5) ( a b )n = an bn, (b 0) Power rule of quotient 6) a 0 = 1, (a 0) Zero power rule 7) 0 0 is not defined. 1 a n = (1 a )n = a n, (a 0) Negative power EXAMPLE 1 (Product rule) YOUR TURN 1 Simplify each expression (all variables are not zero) Simplify each expression (all variables are not zero) (A) x x = x x x 5 factors of x x factors of x factors of x = x + = x 5 (A) t t 5 t 7 (B) x y 5xy = ( 5)(x x)(y y ) = 10x 4 y 5 Commutative and associative laws (B) 5a 5 b 6ab (C) ( 6x y)( x 4 y 5 ) (C) 5x (xy) 0 = 5x 1 Zero power rule = 5x (D) x (x ) 0

3 EXAMPLE (power rule) YOUR TURN Simplify each expression Simplify each expression (A) (x ) (A) ( x ) = ( ) (x ) Power rule = 8x 6 (B) ( x 5 ) 4 (B) ((xy ) 4 ) (C) (5x y) = (xy ) 1 Power rule = x 1 y 4 Power rule (D) ((x y) 4 ) EXAMPLE YOUR TURN Simplify each expression Simplify each expression (A) ( x ) (A) ( x y )4 = x Power rule = x 8 (B) ( 1 x y 5 ) (B) ( 1 x ) = ( x ) = x 6 ( ) Power rule = ( ) x 6 Negative power rule = 4 x 6

4 4 EXAMPLE 4 (Quotient rule) YOUR TURN 4 Simplify each expression with positive exponents. (A) 4x7 y 5 8x 4 y = ( 4 x x x x x x x y y y y y ) 8 x x x x y y = x y Quotient rule Simplify each expression with positive exponents. (A) 1x y 4 x y (B) (x ) (x ) (x ) = x 6 x (x 6 ) = x x 6 x 6 = x 4x 6+6 = x 4x 1 = 4x 10 Power rule Negative power rule Product rule Quotient rule (B) 10x9 (x ) 5 x 10 EXAMPLE 5 YOUR TURN 5 Simplify each expression with positive exponents. (y ) ( y 4 ) Simplify each expression with positive exponents. ( 1 x4 y ) (y ) (4 1 y ) Negataive power rule = y 9 4 y 4 Power rule = 4 y 9+( 4) Product rule = 4 y5 Negataive power rule = 7y5 16

5 5 DEFINITION: A Term is either a single number or a variable, or numbers and variables multiplied together. expression 4x 9y + 5 terms three terms coefficients constant 4x 9y + 5 variable DEFINITION: Terms with exactly same variables that have the same exponents are like terms: 9x, 6x like terms 9x, 6x NOT like term COMBINING LIKE TERMS: Add or subtract the coefficients of the like terms. x 5x + 4 x + x = x x + 4 EXAMPLE 6 YOUR TURN 6 Perform the operations and simplify each expression. Perform the operations and simplify each expression. (A) (x y) (4x y) (A) (5x 7x + 11) (x 4x ) = 5x 7x + 11 x + 8x + 6 = x + x + 17 (B) x(x y) y(5x y) (B) x(x y) 7y(4x y) = x xy 5xy + y = x 7xy + y (C) ( x) 4( x) + 7 = ( x)( x) 4x + 8x + 7 = (4x ) + 8x + 7 (C) 5(x) (x) + 11 = 1x + 8x + 7

6 6 EXAMPLE 7 YOUR TURN 7 Expand each expression and simplify Expand each expression and simplify (A) (x y)(4x + 5y) (A) (x ) = (x )(x ) = 4x 6x 6x + 9 FOIL = 4x 1x + 9 (B) (5x y) (B) (x ) = (x )(x )(x ) FOIL = (9x 1x + 4)(x ) Distributive law = 7x 18x 6x + 4x + 1x 8 (C) (x 1) = 7x 54x + 6x 8 EXAMPLE 8 YOUR TURN 8 Simplify (x 1) (x 1) + 6 Simplify (x ) 5(x ) + (x 1) (x 1) + 6 = (x 1)(x 1) (x 1) + 6 FOIL = (4x x x + 1) (x 1) + 6 = 8x 4x 4x + 6x Distributive law = 8x 14x + 11

7 7 PRACTICE PROBLEM 1. Perform the operations and simplify each expression. (A) (x + y) 6 (x + y) 4 (B) a a 4 a (A) (x + y) 10 (B) a 9 (C) ( 5) 6 (D) ( x )1 (C) ( 5) ( 5) 4 (D) ( x )7 ( x )5 (E) x ( x) 8 x 7 (F) ( x) ( x) 5 x 11 (E) x 18 (F) x 19. Perform the operations and simplify each expression. (A) 5x y 9 ( xy 11 ) (B) xy 5 z ( 4x 5 yz ) (C) ab c 9a 4 b 5 c 6 (D) 5x y 7 w 11 x 7 y 5 (E) ( 4x y 5 )(xy ) (F) ( 5x y )(xy )( 7x 5 y) (G) x y 4x y 5 ( xy) (H) 5a 5 b(a + b) 6 6ab (a + b) (A) 15x 4 y 0 (B) 8x 6 y 6 z 4 (C) 7a 5 b 7 c 9 (D) 15x 9 y 1 w 11 (E) 1x y 8 (F) 70x 9 y 5 (G) 4x 6 y 7 (H) 0a 6 b (a + b) 9. Perform the operations and simplify each expression. (A) 6y 9y 4xy5 (B) xy (C) 0x5 y 6 (D) 6x6 y 7 10x y 1x 5 y (E) 10x9 (x ) 5 16x4 (x+)5 (F) x 10 1x(x+) 4. Perform the operations and simplify each expression. (A) ( 5) 0 (B) (x + 5) 0 (C) (4x) 0 (D) xy ( xy 5 )0 (A) 4y (B) 1y (C) x y 4 (D) xy 4 (E) 5x 14 (F) 4 x (x + ) 4 (A) 1 (B) 1 (C) (D) xy 5. Perform the operations and simplify each expression. (A) (xy) (B) ( x 5 ) 4 (C) (5x y) (D) ( x 5 ) (E) ((x ) 4 ) (F) [(x y) 4 ] (A) 8x y (B) 16x 0 (C) 5x 4 y (D) 18x 10 (E) x 4 (F) 4(x y) 8 (G) 8 x 6 (G) ( x ) (H) ( x y )4 (I) ( x z (y ) 4) (J) ( 5x4 (x 1) 5 ) y 7 (H) (I) (J) x 1 16y 8 8x 6 z (y ) 1 5x 8 (x 1) 10 9y 14

8 8 6. Simplify and write your answer with only positive exponents. (A) x 4 x 9 (C) x y x 5 y 7 (B) 4 7 (D) 4 x y 4 x y (E) ( xy 4 )(xy 5 ) (F) (x y )( 5x y 5 ) (G) ( x4 y ) (H) (x y) x 6 (y 4 ) 7. Simplify and write your answer with only positive exponents. (A) ( ab) (b c) (B) ( x 4 ) (x ) (C) ( x) 4 (4x y) (E) ( x y )4 ( x4 y ) (D) ( x) (xy ) (F) ( x y )4 ( 6x y ) (G) 1 4 x5 y 8x 4 y (x y) (H) 1 18 x y 5 ( 1 x y ) (x y ) 4 (I) x 18x 4 y 5 1 x 5 y 8. Perform the operations and simplify each expression. (J) 16x y 4x x 4 y 5 (A) (x + x + ) + (x x + 1) (B) ( x + 10x) (x + 9x + 8) (C) (x 4x + ) + ( x + 8x + ) (D) (x y xy + y ) (x y 5y ) 9. Perform the operations and simplify each expression. (A) x(x x 4) (B) (x 5)(x + 4) (C) (x y)(4x 5y) (D) (y )(y + ) (E) 5(x 8y) + (5x + 6y) (G) (5x y) (H) (xy 5) 10. Perform the operations and simplify each expression. (F) x(x y) 7y(4x y) (A) (a + b)(a ab + 9b ) (B) (x y)(x + xy + y ) (C) x(x ) (D) (x 4y) (E) (x )(x + 1) (F) (x ) 11. Perform the operations and simplify each expression. (A) {x (y )} (C) {x (x y)} + 4x (B) x {4x (x y)} (D) {(x y) + 5x} (x y) (A) 1 x 5 (B) (C) (D) y 9 x 8 x 5 16y 6 (E) 6x y (F) 10x5 y (G) 9x 8 y 4 (H) 4y 14 (A) 4a b 8 c (B) 7x 6 (C) 4y x 10 (D) 108x 5 y 4 (E) (F) (G) 1 x 4 y 6 8x y 9 x 5 y (H) 8x 10 y 7 (I) 54x 4 y (J) 8x 8 y 4 (A) 4x x + 4 (B) x + x 8 (C) x + 4x + 5 (D) x y xy + 8y (A) x + 6x + 8x (B) 6x 7x 0 (C) 1x xy + 10y (D) 9y 4 (E) 5x + 58y (F) x 0xy + 1y (G) 5x 0xy + 9y (H) x y 10xy + 5 (A) a + 7b (B) x + x y + 4xy y (C) x 1x + 1x (D) 18x + 48xy y (E) x + x x (F) 8x 6x + 54x 7 (A) x + y (B) x y (C) x + 4y (D) x y

9 9 LECTURE 1- RADICALS (TEXTBOOK A.10) Video 1) Simplifying radical expressions ) Rationalizing a denominator PRINCIPAL n-th ROOT: For a positive integer n > 1, n the principal n-th root of a, denoted by a, is a number b such that n a = b means that a = b n where a is the radicand and n is the index. NOTE: For any real number a n a n a, if n is even integer = { a, if n is odd integer When n is even, n a When n is odd, n a is not real number if a < 0 n = a EXAMPLE 1 YOUR TURN 1 Evaluate each expression Evaluate each expression (A) 100 = 10 = 10 = 10 (A) 81 (B) = 5 First, simplify the inside = 5 = 5 (C) ( 7) = 7 = 7 (B) (C) ( 5) EXAMPLE YOUR TURN 4 Evaluate 16 6 Evaluate 1 4 x = 16 means x 4 = 16 x is not a real number since it is impossible that x 4 < 0.

10 10 EXAMPLE YOUR TURN Evaluate each expression 5 (A) 5 = 5 = Evaluate each expression (A) 8 (B) 7 = ( ) = (B) 64 PROPERITES: Let n be an integer and let A, B be real numbers. ab = a b a b = a b (A a)(b b) = A B ab EXAMPLE 4 YOUR TURN 4 Simplify each expression Simplify each expression (A) 8 (A) 1 = 4 = 4 = (B) 18 (B) 8 = 9 = ( 9 ) (C) 45 = ( ) = 9 (D) 75 EXAMPLE 5 YOUR TURN 5 Simplify 16 Simplify = = =

11 11 EXAMPLE 6 YOUR TURN 6 Simplify Simplify (A) = = 5 = (B) ADDING AND SUBTRACTING RADICAL EXPRESSIONS: Adding and subtracting radical expressions is like adding and subtracting like terms: + 4 = ( + 4) = 7 EXAMPLE 7 YOUR TURN 7 Simplify 75 1 Simplify (A) = 5 4 = (5 ) ( ) = 15 4 = 11 (B)

12 1 EXAMPLE 8 YOUR TURN 8 Simplify ( 5) Simplify each expression (A) (5 )(5 + ) ( 5) = ( 5)( 5) = FOIL = Simplify = (B) (4 7) DEFINITION OF CONJUGATE: The conjugate of a + b m is a b m The conjugate of a n + b m is a n b m RATIONALIZATION: A fraction that contains a radical in its denominator can be written as an equivalent fraction with a rational denominator (a denominator without a radical): Multiply the conjugate of its denominator. EXAMPLE 9 YOUR TURN 9 Rationalize the denominator: 5 Rationalize the denominator: 5 5 = 5 = 5 = 5 6 Multiple by ( ) =

13 1 EXAMPLE 10 YOUR TURN 10 Rationalize the denominator: Rationalize the denominator: (5 10) = (5 + 10)(5 10) Multiple conjugate of the denominator (5 + 10) = FOIL = = (5 10) EXAMPLE 11 YOUR TURN 11 Rationalize the denominator: 7 Rationalize the denominator: ( 7 + ) = ( 7 )( 7 + ) ( 7 + ) = Multiple conjugate of the denominator FOIL = = = ( 7 + ) 7 ( 7 + ) 4 7 +

14 14 PROPERITES: Let n and m be integers. a = a = a 1 n a m m n = a m n a mn = a EXAMPLE 1 YOUR TURN 1 Simplify each expression by using radical expressions. (Assume that all variables are positive) (A) 8 / = 8 = 64 = 4 = 4 Simplify each expression by using radical expressions. (Assume that all variables are positive) (A) 7 1 (B) ( 16x6 ) 9 = 16x6 9 = 16 (x ) 9 = 4x Power rule (B) (8x 6 y 9 ) 4 EXAMPLE 1 YOUR TURN 1 Rewrite the expression using rational exponents. Assume that x > 0. (A) x = x Rewrite the expression using rational exponents. Assume that x > 0. (A) x 5 (B) x = ( x ) 1 = (x ) 1 = x 1 8 (B) x 4 (C) x

15 15 PRACTICE PROBLEMS 1. Evaluate (A) 9 (B) 64 (C) 0.04 (D) (E) (G) 8 6 (I) 64 (F) 9 (H) 15 5 (J) 4 (A) (B) 8 (C) 0. (D) 1 (E) 7 (F) Not real number (G) (H) 5 (I) (J). Simplify the radical expression. (A) 18 (C) 196 (E) 75 (G) 81 (I) 48 (B) 8 (D) 80 (F) 45 4 (H) 6 (J) 1 (A) 8 (B) 7 (C) 8 (D) 4 5 (E) 10 (F) 9 5 (G) 4 (H) (I) (J) 6 Not real number. Simplify the radical expression: (A) 6 8 (C) (E) (B) (D) (F) (G) 4 (H) (I) (J) 9 6 (A) (B) (C) (D) (E) (F) (G) (H) (I) 10 (J) 4. Simplify the radical expression. (A) (C) (B) (D) (A) 10 5 (B) 6 (C) 14 (D) 0 (E) 11 5x (E) 5 80x 45x (F) 6 18x 18x (G) (H) 5( 40 10) (I) (5 6)(5 + 6) (J) ( 7 ) (F) 6 x (G) 0 (H) 15 (I) 19 (J) 9 14

16 16 5. Rationalize the denominator: (A) 5 (C) (E) (G) a b a+ b (B) + (D) + (F) + (H) x 9 x+ (A) 5 6 (B) (C) 5 (D) 7 4 (E) (F) (G) ( 10+ 7) 5 6 a+b ab a b (H) x 6. Simplify the radical expressions. Assume that all variables are positive. (A) 5 6 ( ) (B) 7 0 ( ) (C) (E) (1 ) + 1 (G) (I) 4 16 (D) +4 ( 6 ) (F) 1 8 ( 5 ) + (H) (J) Rewrite the expression using exponential form and simplify. (A) 5 1 (B) ( 8) 1 (C) 16 (D) ( ) 5 8. Rewrite each expression with rational exponents. Assume that all variables are positive. (A) ( ( 4) (C) x x (E) 1 x x 9 (G) ( x 6 ) (B) ( 4 6 ) (D) x x (F) 4 y ) 6 (H) 1 x 1 6 x 9. Simplify and rewrite this using radical form. Assume that all variables are positive. (A) (8x) 1/ (B) 5x 1/ (C) 9x (D) (9x) 4 (E) x x 4 (F) x (A) (B) (C) (D) 4 (E) 6 + (F) (G) 6 (H) 6 (I) 4 (J) (A) 5 (B) (C) (D) 8 (A) 16 (B) (C) x (D) x 7 (E) x 5 (F) x (G) x 4 y (H) x 1 (A) x (B) 5 x (C) 9x x (D) 7x x (E) (F) 4 x 8 x

17 17 LECTURE 1- COMPLEX NUMBERS (TEXTBOOK A.7) Video 1) Complex numbers ) Set up your calculator to calculate complex numbers COMPLEX NUMBERS: (Real numbers are complex numbers) Imaginary Unit: i = 1 and i = 1 Standard Form of Complex Numbers: a + bi where a, b are real numbers a + b i real part imaginary part The conjugate of z is z = a + bi = a bi If two complex numbers are equal; x + yi = a + bi x = a and y = b ADDITION AND SUBTRACTION: Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) (c + di) = (a c) + (b d)i EXAMPLE 1 YOUR TURN 1 Find the values of x and y Find the values of x and y x + (y )i = + xi 4 + (x + y)i = x + i Since a + bi = c + di means a = c and b = d, x = ; y = x x = ; y = x + = + = 5

18 18 EXAMPLE YOUR TURN Write 9 in the standard form. Write the following numbers in the standard form. (A) 16 9 = 9 1 = i (B) 49 EXAMPLE YOUR TURN Write each expression in the standard form. Write the following numbers in the standard form. (A) 4 (A) 18 = 9 1 = i (B) (B) 4 9 = (i)(i) Change it in standard form = 6i (C) 44 = 6( 1) i = 1 = 6 CAUTION: 4 9 ( 4)( 9) (D) 9 16

19 19 EXAMPLE 4 YOUR TURN 4 Simplify the expression and write your answer in standard Simplify the expression and write your answer in standard form; ( + i) 4(1 i) form. (A) 5( i) ( 7i) ( + i) 4(1 i) = 6 + i 4 + 8i By distributive law = + 10i (B) ( 5i) + (8 + 6i) EXAMPLE 5 YOUR TURN 5 Simplify the expression and write your answer in standard Simplify the expression and write your answer in standard form: form: ( i) (A) ( i)( + i) ( i) = ( i)( i) = 4 6i 6i + 9i FOIL (B) ( 5i)(5 i) = 4 6i 6i + 9( 1) i = 1 = 4 6i 6i 9 = 5 1i (C) (4 + i)

20 0 NOTE: Since i 4 = (i ) = ( 1) = 1, for any integer n i 4n+1 = (i 4 ) n 1 i 4n+ = (i 4 ) n 1 i = i i = 1 1 i 4n+ = i 4n+ i = i i 4n = (i 4 ) n = 1 DEFINITION: The conjugate of z = a + bi, denoted by z, is z = a + bi = a bi where a and b are real numbers. EXAMPLE 6 YOUR TURN 6 Write i 51 in the standard form. Write each expression in the standard form. (A) i 70 Since i 4n = 1 (the power is a multiple of 4) and 51 = 4 1 +, i 5 = i 48 i = (i 4 ) 1 ( i) = 1 1 ( i) = i (B) i 60 EXAMPLE 7 YOUR TURN 7 Write i 15 in the standard form. Write each expression in the standard form. (A) i 1 Since i 4n = 1 (the power is a multiple of 4) and 15 = 4 ( 4) + 1, i 15 = i 16 i = (i 4 ) 4 i = 1 4 i = i (B) i 6 EXAMPLE 8 YOUR TURN 8 Find the conjugate of i Find the conjugate of the following complex number. (A) 5 + 4i The conjugate of i is + i. (B) 7 5i (C) 4i

21 1 EXAMPLE 9 YOUR TURN 9 Simplify it in the standard form: + i = ( i) ( + i)( i) +i Multiple conjugate of the denominator Simplify it in the standard form: 7 4 i = 6 4i FOIL 9 6i + 6i 4i = 6 4i 9 4( 1) i = 1 = 6 4i 1 = i EXAMPLE 10 YOUR TURN 10 Simplify it in the standard form: +i i Simplify it in the standard form: i 5+4i + i i ( + i)( + i) = ( i)( + i) Multiple conjugate of the denominator = 6 + i + i + i FOIL 9 + i i i = 6 + 5i + ( 1) 9 ( 1) i = 1 = 5 + 5i 10 = i = i Your Turn: Use your calculator to rationalize the denominator. (A) +i (B) i i (C) +i 1 i (D) 6 i 4+i

22 PRACTICE PROBLEMS 1. Find the conjugate of the following complex numbers (A) + i (B) 4i + 1 (C) 8 (D) 1i (A) i (B) 1 + 4i (C) 8 (D) 1i (E) 1 i (E) + 1 i (F) 5 7i (F) 5 + 7i. Simplify the complex number in the standard form. (A) i 48 (B) i 54 (C) i 47 (D) i 89 (E) ( + i) + (4 i) (F) (6 i) (4 4i) (G) ( + 7i) + (5 + 4i) (H) ( + 6i) (6 8i) (I) (1 i) + ( i) (J) (1 + i) (4 5i) (K) ( i) + ( + i) (L) ( i) (4 i) (M) 4 (N) 1 9 (O) 8 (P) 1 (A) 1 (B) 1 (C) i (D) i (E) 7 i (F) + i (G) i (H) i (I) 7 11i (J) 9 + 1i (K) 7 (L) 5i (M) i (N) 1 i (O) 4i (P) 6. Simplify the complex number in the standard form. (A) 5i(4 6i) (C) ( + i)(4 i) (E) ( i)( i) (B) 4i( 8i) (D) (6 i)(4 5i) (F) ( i)(1 + i) (G) (5 i)(5 + i) (H) (4 i) (I) (5 + i) (J) (1 + i) 4. Simplify the complex number in the standard form. (A) 1 i (C) 5i 6 5i (E) +i 4 i (G) +i i (I) i 5 (B) +i 1 i (D) +5i (F) i +i (H) i 4 (J) i 14 (A) 0 + 0i (B) 1i (C) 18 i (D) 14 8i (E) 10i (F) (G) 9 10 i (H) 7 4i (I) (J) (A) 1 + 0i i i (B) i (C) i (D) i (E) (F) i i (G) 1 i (H) i (I) (J) i 1

23 5. Simplify the complex number in the standard form. (A) 8 (C) 7 (B) 7 4 (D) 7 (A) 16i (B) 7 (C) i (D) i (E) 0 (F) i (E) (F) Simplify the complex number in the standard form: (A) i + i + i + + i 10 (B) i + i + i + + i 4000 (A) i 1 (B) 0 7. Find the real numbers x, y which satisfy (A) 1 + xi = y + i (B) (x + 1) + (y )i = 0 (C) (x + i)( + 4i) = 11yi + 4 (D) x 1+i + y 1 i = 5 +i (A) x =, y = 1 (B) x = 1, y = (C) x = 4, y = (D) x =, y = 1 (E) x = 15 1 y = 8 8 (F) x = y = 1 (E) x + y = 1+i 1 i i (F) [x(1 + i) y(1 i)] = 1 8. Simplify the complex number in the standard form. (A) ( i) + ( + i) (B) (4 + i) (4 i) (C) ( i) 10 (D) (1 + i) 7 (E) 5 5+ (G) i + 5i 7 (I) + i i +i (F) (H) (+i)(4 i) (1+i) (J) i 4 i 4+i (A) 6 (B) 48i (C) i (D) 8 8i (E) (F) 1 15i i (G) 4i (H) 5 5i (I) (J) 5+4i 5 (K) 1 (L) i 11+10i 5 (K) ( 1+i 1 i )100 (L) ( 1 i 1+i ) Find the real number a if the standard form of i(a + i) is a real number. a = 10. Evaluate (A) x x + 7 if x = 1 i (A) (B) 1 (B) x 6x + 11 if x = + i

24 4 LECTURE 1-4 BASIC FACTORS Video 1) Greatest common Factor ) A B factoring ) Grouping factoring 4) Trinomial factoring 5) Factoring by using substitution GREATEST COMMON FACTOR EXAMPLE 1 YOUR TURN 1 Factor out the greatest common factor: Factor out the greatest common factor. (A) 9x 4 6x + x (A) 10x 5x GCF 5x = 5x (x) + 5x ( 1) = 5x (x 1) (B) 1a b 5 18a b (B) 5x y x yz x y z GCF x y = (x y)(5y ) + (x y)(10xz) (x y)(y z ) = x y(5y + 10xz y z ) (C) 1x 4 y + 8x y 5 4x y EXAMPLE YOUR TURN Factor out the greatest common factor. Factor out the greatest common factor. x(x + ) (x + ) (A) x(x + ) 6(x + ) x(x + ) (x + ) = (x)(x + ) + ( 1)(x + ) GCF (x + ) (B) x (x ) + (x ) = (x + )(x 1)

25 5 EXAMPLE YOUR TURN Factor out the greatest common factor. Factor out the greatest common factor: 4x + 9y 5x We cannot find the greatest common factor of 5x,. This is a prime factor. SPECIAL FACTOR METHOD 1 A B = (A B)(A + B) A + B : Prime factor EXAMPLE 4 YOUR TURN 4 Factor Factor (A) x 5 (A) x 16 = x 4 = (x 4)(x + 4) (B) 16x 49 (B) 4x 9 = (x) = (x )(x + ) (C) 64x 81y EXAMPLE 5 YOUR TURN 5 Factor x + 4. Factor 9x + 1. x + 4 is a prime factor: we cannot factor the expression

26 6 SPECIAL METHOD : GROUPING When the expression consists of 4 terms, we can use grouping (not always) EXAMPLE 6 YOUR TURN 6 Factor completely: 4x 6x y 6xy + 9y Factor completely: (A) 10px 15qx + 8py 1qy 4x 6x y 6xy + 9y = (4x 6x y) (6xy 9y ) GCF x GCF y = x (x y) y (x y) = (x y)(x y ) Factor out x y (B) x 6xy x + 18y EXAMPLE 7 YOUR TURN 7 Factor completely: x x 5x + 50 Factor completely: (A) x 5x 1x + 0 x x 5x + 50 = (x x ) (5x 50) GCF x GCF 5 = x (x ) 5(x ) = (x ) (x 5) A B form = (x )(x 5)(x + 5) Factor out x (B) x + x 18x 7

27 7 TRINOMIAL FACTOR 1: x + bx + c = (x + A)(x + B) where b, c are integers. We must find two integers A and B such that b = A + B and c = AB. EXAMPLE 8 YOUR TURN 8 Factor each trinomial Factor each trinomial (A) x + 7x + 1 = (x + )(x + 4) (A) x + 15x + 44 We must find two integers A, B such that AB = 1, A + B = 7 The numbers are +, +4 (B) x 11x + 18 = (x 9)(x ) We find two integers A, B such that AB = 18, A + B = 11 The numbers are 9, (B) x 7x + 10 (C) x x 8 = (x 4)(x + ) We find two integers A, B such that AB = 8, A + B = The numbers are 4, + (C) x x 4 (D) x + x 5 = (x + 7)(x 5) We find two integers A, B such that AB = 5, A + B = + The numbers are +7, 5 (D) x + 9x 6 (E) 6 x x = (x + x 6) = (x + )(x ) We find two integers A, B such that AB = (1)( 6) = 6, A + B = 1 The numbers are, + (E) 18 x x

28 8 TRINOMIAL FACTOR : ax + bx + c, a 1,0 where a, b, c are integers. We must find two integers A and B such that b = A + B and ac = AB. EXAMPLE 9 YOUR TURN 9 Factor: x + 5x 1 Factor: x + 11x + 6 We must find two integers A, B such that AB = ()( 1) = 4, A + B = 5 The numbers are, +8. So x + 5x 1 = x x + 8x split = x x GCF x 1 + 8x 1 GCF 4 = x(x ) + 4(x ) Grouping = (x )(x + 4) EXAMPLE 10 YOUR TURN 10 Factor: 6x 7x 5 Factor: 8x + x We find two integers A, B such that AB = (6)( 5) = 0, A + B = 7 The numbers are 10, +. So 6x 7x 5 = 6x 10x + x split = 6x 10x GCF x 5 + x 5 GCF 1 = x(x 5) + (x 5) Grouping = (x 5)(x + 1)

29 9 SPECIAL FACTORING 1) A B = (A B)(A + AB + B ) ) A + B = (A + B)(A AB + B ) EXAMPLE 11 YOUR TURN 11 Factor completely: x 8 Factor completely: 8x 15 x 8 = x A B = (x )(x + x + ) = (x )(x + x + 4) EXAMPLE 1 YOUR TURN 1 Factor completely: 7x + 8 Factor completely: 64x + y 7x + 8 = (x A ) + B = (x + )((x) (x) + ) = (x + )(9x 6x + 4)

30 0 HOW TO FACTOR AN EXPRESSION: 1) First, find GCD ) Use a special method. EXAMPLE 1 YOUR TURN 1 Factor completely: Factor completely: (A) x 18x (A) 6x 4x = 4x(9x 1) Find GCF = 4x((x) 1 ) Use A B = 4x(x 1)(x + 1) (B) x x 1x (B) x 8x + 1x = x(x 8x + 1) Find GCF = x(x )(x 6) EXAMPLE 14 YOUR TURN 14 Factor completely: x 4 16 Factor completely: x 4 81 x 4 16 = (x ) 4 Use A B = (x 4) (x + 4) factor again = (x )(x + 4) Use A B = (x )(x + )(x + 4)

31 1 PRACTICE PROBLEMS 1. Factor completely by using greatest common Factor (A) x y + xy + 4xy (B) 18x y 5 1x y + 0x y (C) 4x y z 6x y z (D) 4x y + 6xy 1x y (E) a(5m 4) + b(5m 4) (F) x (y + 1) 5(y + 1) (G) a(b + ) + c(b + ) (H) (x y)a + (y x) (A) xy(x + y + 4) (B) 6x y (y y + 5) (C) x y z(y xz) (D) xy(xy + y 6x ) (E) (5m 4)(a + b) (F) (x 5)(y + 1) (G) (b + )(a + c) (H) (x y)(a 1). Factor completely. Write prime if it is a prime factor. (A) x + 1x + 7 (B) x + 1x + 54 (C) x 15x + 6 (D) x 1x + 6 (E) x + 4x 1 (F) x 9x 6 (G) x + 1x 6 (H) x + x 56 (I) x 16 (J) x 6 (K) x 9x 6 (L) x x 75. Factor completely. Write prime if it is a prime factor. (A) x 5x + (B) x x 14 (C) x + 10x 8 (D) x 8x + 4 (E) 5x + x + 7 (F) 5x + 1x + 4 (G) 7x 11x + 4 (H) 7x + 16x Factor completely. Write prime if it is a prime factor. (A) 10x + 1x 0 (B) 1x + 7x + 1 (C) 4x + 8x + (D) 8x 10x + (E) 6x 7x 0 (F) 9x + 6x + 1 (G) 14x + 4x + 1 (H) 10x 9x + 5. Factor completely. Write prime if it is a prime factor. (A) 15x + 18x + 0x + 4 (B) 8x x 1x + (C) x xy ax + ay (D) xy + x + 9y + 7 (A) (x + )(x + 9) (B) (x + )(x + 18) (C) (x )(x 1) (D) (x 4)(x 9) (E) (x )(x + 7) (F) (x 1)(x + ) (G) Prime (H) (x 7)(x + 8) (I) (x 4)(x + 4) (J) (x 6)(x + 6) (K) (x 1)(x + ) (L) (x 5)(x + ) (A) (x )(x 1) (B) (x 7)(x + ) (C) (x )(x + 4) (D) (x )(x ) (E) Prime (F) (5x + )(x + ) (G) (7x 4)(x 1) (H) (7x 5)(x + ) (A) (x + 5)(5x 6) (B) (x + 1)(4x + 1) (C) (x + )(x + 1) (D) (x 1)(4x ) (E) (x 5)(x + 4) (F) (x + 1) (G) Prime (H) (x 1)(5x ) (A) (5x + 6)(x + 4) (B) (x )(4x 1) (C) (x y)(x a) (D) (x + 9)(y + )

32 6. Factor completely. Write prime if it is a prime factor. (A) 4x 9 (B) 9x + 16 (C) 16x y (D) 7x 1y (E) 16x 5 49 (F) 9x 1 4 y (G) (x + y) 5 (H) (x ) (y + 1) 7. Factor completely. Write prime if it is a prime factor. (A) x x 5x + 50 (B) x + x 4x 1 (C) x 4x 1x + 16 (D) x 5x 8xy + 0y 8. Factor completely. Write prime if it is a prime factor. (A) 8x + 7y (B) 64x y (C) 64x + 7y (D) x 8y (E) x y 7 (F) 8y Factor completely. Write prime if it is a prime factor. (A) x 8x (B) x y 75y (C) xy x y (D) x 4 16 (E) a 4 b 4 (F) x 6 y 6 (A) (x )(x + ) (B) Prime (C) (4x y)(4x + y) (D) (x y)(x + y) (E) (4x 5 7 ) (4x ) (F) (x y ) (x + y ) (G) (x + y 5)(x + y + 5) (H) (x y 4)(x + y ) (A) (x 5)(x + 5)(x ) (B) (x )(x + )(x + ) (C) (x 4)(x )(x + ) (D) (x 5)(x y)( x + y) (A) (x + y)(4x + 6xy + 9y ) (B) (4x y)(16x + 4xy + y ) (C) (4x + y)(16x 1xy + 9y ) (D) (x y)(x + xy + 4y ) (E) (xy )(x y + xy + 9) (A) x(x )(x + ) (B) y(x 5)(x + 5) (C) xy(y x)(y + x) (D) (x + 4)(x )(x + ) (E) (a b)(a + b)(a + b ) (F) (x y)(x + y)(x xy + y )(x + xy + y ) 10. Factor completely. Write prime if it is a prime factor. (A) (x + 1) + (x + 1) 10 (B) (x + ) 9(x + ) + 18 (C) (x x) x + 6x 8 (D) (x x)(x x + 5) + 6 (E) (x x)(x x + 1) 6 (F) (x y)(x y ) + (A) (x + 6)(x 1) (B) (x 1)(x 4) (C) (x 4)(x )(x 1)(x + 1) (D) (x 1)(x )(x x + ) (E) (x + 1)(x )(x x + ) (F) (x y 1)(x y ) 11. Factor completely. Write prime if it is a prime factor. (A) x 4 10x + 9 (B) x 4 6x 7 (C) x 4 4x y + 6y 4 (D) x 4 + 4y 4 (E) x 4 + 7x + 16 (F) x 4 + x + 5 (A) (x 1)(x + 1)(x )(x + ) (B) (x + )(x )(x + ) (C) (x 4xy + 6y ) (x + 4xy + 6y ) (D) (x xy + y ) (x + xy + y ) (E) (x + x + 4)(x x + 4) (F) (x x + 5)(x + x + 5)

33 LECTURE 1-5. SOLVING THE EQUATIONS (TEXTBOOK A.6) Video 1) Solving linear equations ) Square root method to solve a quadratic equation ) Quadratic formula to solve a quadratic equation 4) Solving higher degree equation by factoring 5) How to program quadratic formula in TI-8: LINEAR EQUATIONS: Ax + B = 0 Ax = B x = B A EXAMPLE 1 YOUR TURN 1 Solve the equation: (x 4) + 5 = (x 1) Solve the equation: 4(x + 5) (x 8) = 9 x = 6x x 6x = Distributive law Add 6x 4x = 4x = + Add 8 4x = 0 x = 0 Divide by 4 EXAMPLE YOUR TURN Solve the equation: x 1 5 = Solve the equation: 4x 7 = x = 5 Multiply 5 x 1 = 15 x = 16 Add 1 x = 16 = 8

34 4 QUADRATIC EQUATIONS: Solve ax + bx + c = 0, a 0 Square root method: when b = 0, x = c a x = ± c a By factoring: when you can find factors. Quadratic formula (always you can find the solutions): x = b ± b 4ac a EXAMPLE YOUR TURN Solve the equation: 49x = 1 Solve the equation (real or complex solutions): (A) x = 45 49x = 1 x = 1 49 No x Term Square root method x = ± 1 49 = ± 4 = ± 7 7 (B) 6x 7 = 0 (C) 5x = 9 EXAMPLE 4 YOUR TURN 4 Solve the equation: x x = 1 Solve the equation: x 8 = x x x 1 = 0 Make one side 0 (x 4)(x + ) = 0 Factor x = 4,

35 5 EXAMPLE 5 YOUR TURN 5 Solve the equation: 6x + 1x 5 = 0 Solve the equation: x + 1x 10 = 0 Since a = 6, b = 1, c = 5, x = (1) ± (1) 4 6 ( 5) (6) 1 ± 89 = 1 1 ± 17 = 1 Therefore, x = 4 0, or x = 1 1 x = 1, 5 EXAMPLE 6 YOUR TURN 6 Solve the equation: x + 1 = 4x Solve the equation: x + 1 = x x 4x + 1 = 0 Make one side 0 Since a =, b = 4, c = 1, x = ( 4) ± ( 4) 4 1 () = 4 ± 8 4 = 4 ± 4 = 4 4 ± 4 = 1 ±

36 6 EXAMPLE 7 YOUR TURN 7 Solve the equation: x + 1x + 1 = 0 Solve the equation: 4x 4x + 1 = 0 Since a =, b = 1, c = 1, x = 1 ± (1) 4 1 () 1 ± 0 = 6 = 1 6 = EXAMPLE 8 YOUR TURN 8 Solve the equation: 4x = 4x 7 Solve the equation: x + 1 = 4x 4x 4x + 7 = 0 Make one side 0 Since a = 4, b = 4, c = 7, x = ( 4) ± ( 4) (4) = 4 ± 96 8 = 4 ± 4 6 i 8 = 4 8 ± 4 6 i 8 = 1 ± 6 i

37 7 HIGHER ORDER POLYNOMIAL: To find solutions, 1) Make one side zero ) Factor as the product of linear factors or quadratic factors EXAMPLE 9 YOUR TURN 9 Solve the equation: (A) x x = 0 x (x ) = 0 x = 0, x = 0, or x = double solution x = 0, The solution set is {0,} Factor out GCF Solve the equation: (A) x = 7x (B) x = 18x (B) 5x 4 0x = 0 5x (x 4) = 0 Factor out GCF A B 5x (x )(x + ) = 0 x = 0, x = or x = The solution set is {,0,} EXAMPLE 10 YOUR TURN 10 Solve the equation: x 5x x = 0 Solve the equation: x 7x + x = 0 x 5x x = 0 x(x 5x ) = 0 Factor out GCF x = 0, x 5x = 0 quadratic formula x = 0, x = 0, x = 5 ± 7 4 x = ( 5) ± ( 5) 4 ( ) () x = 0,, 1 The solution set is {0,, 1 }

38 8 EXAMPLE 11 YOUR TURN 11-1 Solve the equation: x x 4x + 8 = 0 Solve the equation: x + 5x 7x 45 = 0 x x 4x + 8 = 0 x (x ) 4(x ) = 0 Grouping (x ) (x 4) A B = 0 (x )(x )(x + ) = 0 x =, x = The solution set is {, }. YOUR TURN 11- YOUR TURN 11- Solve the equation: x x 8x + 1 = 0 Solve the equation: x + x 0x 40 = 0

39 9 PRACTICE PROBLEMS 1. Solve the equation (A) (x ) = 4(x + 5) (B) (4x 1) = (5x ) (C) 5(x 1) = (x 1) (D) 5 (x + 1) = (5 x) (E) 0.(x + ) = 0.4(8 x) (F) 0.0(x 8) = 0.x + 0. (A) 5. (B) (C) (D) 4 (E) 14 5 (F) 9 7. Solve the equation (A) x+1 4 = (x + 1) + 1 (B) x+1 8 (C) 1 6 (x ) = 1 4 x + 4 = x+7 6 (D) 1 x 4 = 0.5x (E) x 1 + x+ = 1 (F) 1 x 7 x 0.x = 5 (G) x+1 = 5x 1 6 (H) 1 x+5 x = + 1 (A) (B) 5 (C) 0 (D) 0 (E) 1 5 (F) 1 4 (G) (H) 16. Solve the equation: (A) x = 4 (A) 5x ( x) = 6 ( x 1) (B) + (5 + 4x) = 9 (4 x) (B) x = 5 4. Find all real/complex solution(s) of the equation: (A) x + 7x = 0 (B) x = 9x (C) 4x + 5x = 0 (D) x = x (E) x = 9x (F) 5x x = 0 (A) 0, 7 (B) 0,9 (C) 0, 5 4 (D) 0, (E) 0, (F) 0, Find all real/complex solution(s) of the equation: (A) x 7x + 10 = 0 (B) x 6x + 9 = 0 (C) x + 1x 10 = 0 (D) 4x + 11x = 0 (E) x x + 1 = 0 (F) x + x 6 = 0 (A) x =,5 (B) x = (C) x =, 5 (D) x = 1, 4 (E) x = 1, 1 (F) x =, 6. Find all real/complex solution(s) of the equation: (A) x = 6 (B) 4x = 6 (C) 4x + 5 = 0 (D) 5x 16 = 0 (E) (x + 4) = 9 (F) (x + ) = 7 (A) x = ±6 (B) x = ± (C) x = ± 5 i (D) x = ± 4 5 (E) x = 1, 7 (F) x = ± (G) (x 1) 5 = 0 (H) (x ) = 49 (G) x = 1± 5 (H) x = ± 7i

40 40 7. Find all real/complex solution(s) of the equation: (A) x 7x + = 0 (B) x + 5x = (C) x + 6x + 1 = 0 (D) x x 7 = 0 (E) x + x = 6 (F) x + 8x = (G) x x 4 = 0 (H) 9x + 6x 4 = 0 (I) x 6x + 1 = 0 (J) x x + 17 = 0 (K) x x + 5 = 0 (L) x 5x + 4 = 0 (M) 4x x + 1 = 0 (N) x x + 1 = 0 (A) (B) 1, 1, (C) ± (D) 1± (E) 1 ± 7 (F) (G) (H) (I) (J) 4± ± ± 5 ± i 1 ± 4i (K) 1 ± i (L) 5±i 7 4 (M) 1±i 4 (N) 1±i 8. Find all real/complex solution(s) of the equation: (A) x(x + 1) = x + 4 (B) x(x + 1) x = 0 (C) (x 1)(x + ) = 4 (D) x(x 6) = (E) x + 8 = 0 (F) x 7 = 0 (G) x 4 = 16 (H) x + 4x + 5x = 0 (A) ± (B) 0, 1 (C), (D) ± (E),1 ± i (F), ± i (G) ±, ±i (H) 0, ± i 9. Find all real/complex solution(s) of the equation. (A) 5x 0x = 0 (B) x = 6x (C) x = 18x (D) 4x 4 8x = Find all real/complex solution(s) of the equation (A) x + x 16x + 48 = 0 (B) x x + 4x 8 = 0 (C) x x 9x + 9 = 0 (D) x + x 4x 1 = 0 (A) x = 0,, (B) x = 0, (C) x = 0,, (D) x = 0, (A) x =,4, 4 (B) x =, ±i (C) x = 1,, (D) x =,,

41 41 LECTURE 1-6. SOLVE THE OTHER EQUATIONS Video 1) Solving higher degree equation by factoring ) Solving radical equations ) Solving rational equations EXAMPLE 1 YOUR TURN 1 Solve the equation: (x + 4) + 4(x + 4) 1 = 0 Solve the equation: (A) (x ) (x ) + = 0 Let A = x + 4 (x + 4) A + 4 (x + 4) 1 = 0 A A + 4A 1 = 0 (A + 6)(A ) = 0 (x )(x + 4 ) = 0 (x + 10)(x + ) = 0 x = 10 or x = (B) (x + ) (x + ) 1 = 0 The solution set is { 10, }

42 4 EXAMPLE YOUR TURN Solve the equation: x 4 5x + 4 = 0 Solve the equation: (A) x 4 1x + 6 = 0 Let A = x x 4 5x + 4 = 0 (x ) 5 (x ) + 4 = 0 A A A 5A + 4 = 0 (A 4)(A 1) = 0 (x 4) (x 1) = 0 A B A B (x )(x + )(x 1)(x + 1) = 0 (B) x x 16 = 0 x =,, 1, 1 The solution set is {, 1,1,} EXAMPLE YOUR TURN Solve the equation: 1 x x 15 = 0 Solve the equation: 1 x x 8 = 0 Let A = 1 x 1 x x 15 = 0 ( 1 x ) ( 1 x ) 15 = 0 A A 15 = 0 (A 5)(A + ) = 0 A = 5 or A = 1 x = 5 or 1 x = x = 1 5, 1 The solution set is { 1 5, 1 }

43 4 RATIONAL EQUATIONS 1) A = C B D means AD = BC ) To solve this equation, we can multiply both sides by the least common denominator: ) Check your answers. EXAMPLE 4 YOUR TURN 4 Solve the equation: x+5 x 5 = 10 Solve the equation: 6 x x = x Since the denominator cannot be zero, x 5 x+5 = 10 x 5 1 (x + 5) 1 = (x 5) 10 x + 5 = 10x 50 9x + 5 = 50 9x = 55 x = 55 = Therefore, the solution set is { 55 9 } Add 10x Add 5 Check that the value is not equal to the excluded value EXAMPLE 5 YOUR TURN 5 Solve the equation: x+4 = x 1+x Solve the equation: x x = x x 1 Since the denominators cannot be zero, x 4, 1 x+4 = x 1+x (1 + x) = x(x + 4) + 6x = x + 4x 0 = x x Add 6x 0 = (x )(x + 1) x =, 1 Therefore, the solution set is {, 1} Check that the values are not equal to any excluded values

44 44 EXAMPLE 6 YOUR TURN 6 Solve the equation 1 x x x 4 = 8 x 16 Solve the equation x x + 4 x + = 18 x 9 First, find the least common denominator. 1 x x x 4 = 8 (x + 4)(x 4) Since the denominators cannot be zero, x 4, 4 Multiply by the LCD (x + 4)(x 4) (x + 4)(x 4) x + 4 x(x + 4)(x 4) 8(x + 4)(x 4) + = x 4 (x + 4)(x 4) x 4 + x(x + 4) = 8 x + 5x 4 = 8 x + 5x + 4 = 0 (x + 4)(x + 1) = 0 x = 4, 1 Since x 4, 4, We have only one solution x = 1. Therefore, the solution set is { 1} Check your answer x Left side: Right side:x = undefined = = undefined 8 ()( 5) = 8 15

45 45 RADICAL EQUATIONS: 4) When you want to solve an equation containing a radical expression, you must isolate the radical on one side from all other terms and then square both sides of the equation. 5) Check your answers. EXAMPLE 7 YOUR TURN 7 Solve the equation: x = 6 Solve the equation: x 1 = x = 6, x 0 x = 6 = ( x) = x = 4 Check your answer Check: x Left side: x Right side: 6 Solution? 4 4 = 6 6 Yes Therefore, the solution set is {4}. EXAMPLE 8 YOUR TURN 8 Solve the equation: 8 x = x Solve the equation: 6 x = x ( 8 x) = x 8 x = x 0 = x + x 8 0 = (x + 4)(x ) x = 4, are possible solutions x Left side: 8 x Right side: x solution 4 8 ( 4) = 16 = 4 4 No 8 () = 4 = Yes Therefore, the solution set is {}.

46 46 EXAMPLE 9 YOUR TURN 9 Solve the equation: 5x + 11 = x + Solve the equation: x + 10 = x + 4 5x + 11 = x + ( 5x + 11) = (x + ) 5x + 11 = x + 6x = x + x 0 = (x + )(x 1) x =, 1 are possible solutions x Left side: 5x + 11 Right side:x + Solution 5( ) + 11 = 1 = 1 + = 1 Yes 1 5(1) + 11 = 16 = = 4 Yes Therefore, the solution set is {, 1}. EXAMPLE 10 YOUR TURN 10 Solve the equation: x = Solve the equation: x + 1 = x = ( x ) = () x = 8 x = 11 is a possible solution. x Left side: x Right side: Solution = 8 = Yes Therefore, the solution set is {11}.

47 47 EXAMPLE 11 YOUR TURN 11 Solve the equation: Solve the equation: x x + = 1 x + 6 x = Isolate one radical on one side of the equation x = 1 + x + ( x ) = (1 + x + ) Add x + Square each side x = 1 + x + + x + x = x x + Subtract x + 4 x 6 = x + Divide by x = x + Square each side (x ) = ( x + ) x 6x + 9 = x + x 7x + 6 = 0 Subtract x + Factor (x 1)(x 6) = 0 x = 1, 6 Check the answers x Left side: Right side: Solution x x = 1 1 No = 1 1 Yes Therefore, the solution set is {6}

48 48 PRACTICE PROBLEMS 1. Find all real/complex solution(s) of the equation (A) x + x = (x + 1)(x + 6) (B) (x 5) 4(x 5) 1 = 0 (C) (x + ) (x + ) 8 = 0 (D) (x ) (x ) 1 = 0 (E) (x + x) 11(x + x) = 4 (F) (x x) + (x x) 1 = 0 (G) (x 1) (x 1) = (H) (x + x) 8(x + x) = 0 (I) x 4 6x + 5 = 0 (J) x 4 8x 9 = 0 (A) x = 1,, (B) x = 1, (C) x = 4 (D) x = 7,0 (E) 4,,,1 (F), 1,1 ± i (G) ±, 0 (H) 5,, 1, (I) ±5, ±1 (J) ±, ±i. Solve the equations: (A) 1 x+1 = 5 (C) x 1 = x 6 x 5 x+ (E) x (B) = x (D) x+1 x = x x+ + x = (F) 5 = x x 4 x 15 x+6 x 9 (G) x x+ + x x 1 = 7 (H) x + 5 x+ = 1 x 4 (A) x = 5 (B) x = (C) x =,16 (D) x = 1, 4 (E) x =,6 (F) x = 9 (G) x = 7, (H) No solution (I) x + = 5 x 1 x x (K) x x 4 4 x+5 = 6 x +x 0 (J) (L) x + 6 = x 8 x 6x x 10x x+1 = 5 x +5x+4 x +x 4 x 1 (I) x = 1 (J) x = 6 (K) No solution (L) x = 6, 4. Solve the equations: (A) x = x + 8 (B) x = 6 5x (C) x + = x + x + 1 (D) x + = x + x + 4 (E) x + 10 x = (F) x + 7 x = (G) x 5 = x + 6 (H) x + 1 = x (I) x = 4x 1 (J) x + 5 x = (K) x + 4 x + 1 = 1 (L) (x 1) = 4 (M) x 5 = x Find all real/complex solution(s) of the equation (N) x = x (A) x x = 0 (B) x x 1 = 6 (C) 1 x 1 x 1 = 0 (D) 1 x 8 x = 0 (A) x = 4 (B) x = 1 (C) x = 1 (D) x = 5 (E) x = (F) x = 1, (G) x = 11 (H) x = 1 (I) x = 1 (J) x = 4 (K) x = 1,0 (L) x = 7,9 (M) x = 6 (N) x = 1, (A) 9 (B) 7, 8 1 (C), (D), 1 10

49 49 LECTURE 1-7 DISTANCE, MIDPOINT, AND CIRCLES Video 1) Find the midpoint and distance between two points ) Introduction to a circle ) Find the radius and its center of the given circle equation 4) Find the circle equation with radius and a point 5) Find the circle equation with two diameter points DISTANCE AND MIDPOINT: Let P(x 1, y 1 ) and Q(x, y ) be two points in the plane. The distance between any two points is d(p, Q) = (x x 1 ) + (y y 1 ) THE INTERCEPT: x-intercept: the x-values at which the graph intersect x-axis y-intercept: the y-values at which the graph intersect yaxis The midpoint M of two points is M = ( x 1 + x, y 1 + y ) (x,y ) 4 (0,) : y- intercept distance midpoint y y 1 x-intercept : (,0) 5 (x 1,y 1 ) x x 1 EXAMPLE 1 YOUR TURN 1 Find the distance and midpoint between two points. Find the distance and midpoint between two points. (, 4) and (5, 4) ( 5, ) and (1, ) Distance = (5 ) + (4 ( 4)) = = 68 = 17 Midpoint = ( +5, ( 4)+4 ) = (4,0)

50 50 CIRCLES: a circle with center ( h, k ) and radius r. The standard form is (x h) + (y k) = r The general form is x + y + ax + by + c = 0 center (h,k) radius r EXAMPLE YOUR TURN Find the center and radius of a circle. Find the center and radius of a circle. (x + 5) + (y 4) = 56 (A) (x ) + (y + ) = 8 Center: ( 5, 4) Radius: 56 = 14 (B) x + (y + 7) = 1 EXAMPLE YOUR TURN Find the center and radius of a circle. Find the center and radius of a circle. x + y 16x + 1y + 6 = 0 (A) x + y 6x + y + = 0 x + y 16x + 1y + 6 divide by x + y 8x + 6y + 1 = 0 = 0 (x 8x) + (y + 6y) = 1 (x 8 half:4 x + 4 ) + (y + 6y) = (x 8x + 16) + (y + 6 half: y + ) = + (B) x + y 4x 1y 1 = 0 (x 8x + 16) + (y + 6y + 9) = 1 (x 4) + (y + ) = 1 Center: (4, ) Radius: 1 =

51 51 EXAMPLE 4 YOUR TURN 4 Find the equation of a circle that Find the equation of a circle that Radius : and Center: (0, ) Radius : 5 and Center: ( 4,) (x h) + (y k) = r (x 0) + (y ( )) = x + (y + ) = 9 EXAMPLE 5 YOUR TURN 5 Find the equation of a circle that Find the equation of a circle that Center: origin, containing (,). (A) Center: (1, ), containing (4, ) To find the equation of the circle, we must calculate the radius (the distance from center and a circle point) : radius is the distance between (0,0) and (,) r = ( 0) + ( 0) = 1 Since the center is (0,0) and radius is r = 1 (x 0) + (y 0) = ( 1) (B) Center:(, ), x-intercept: 4 x + y = 1

52 5 EXAMPLE 6 YOUR TURN 6 Find the equation of a circle such that Find the equation of a circle such that 6 4 O This circle has the center (, 1) and radius (x ) + (y + 1) = 6 (x ) + (y + 1) = 9 EXAMPLE 7 YOUR TURN 7 Find the equation of a circle that the center is (,4) and it Find the equation of a circle that the center is (,) and it is tangent to x-axis. is tangent to y-axis. 1) Draw the graph of the circle 6 4 (,4) 5 ) In the graph, we can find the radius: Then, the equation of the circle is (x ) + (y 4) = (x ) + (y 4) = 9

53 5 EXAMPLE 8 YOUR TURN 8 Find the equation of a circle whose two diameter (A) Find the equation of a circle whose two diameter endpoints are (1,4) and (,). endpoints are (,6) and (4,0). 1) The distance between (1,4) and (,) is diameter of the circle (diameter= radius) r = ( 1) + ( 4) = 0 = 5 r = 5 ) The midpoint of those is the center. The center is ( 1+( ), 4+ ) = ( 1,) Then, the equation of the circle is (x + 1) + (y ) = ( 5) (B) Find the equation of a circle whose two diameter endpoints are (,0) and (1,4). (x + 1) + (y ) = 5

54 54 EXAMPLE 9 YOUR TURN 9 Find the point with coordinates of the form (a, a) that is Find the point with coordinates of the form (a, a) that is in the third quadrant and is a distance 5 from P(1,). in the third quadrant and is a distance from P(,1). Since it is in III quadrant, (a, a) = (, ). The distance between P(1,) and (a, a) is 5, (a 1) + (a ) = 5 4a 4a a 6a + 9 = 5 5a 10a + 10 = 5 By taking square to both sides, 5a 10a + 10 = 5 5a 10a 15 = 0 5(a a ) = 0 5(a )(a + 1) = 0 a =, 1 Since a < 0 a = 1 The point is (a, a) = (, 1).

55 55 PRACTICE PROBLEMS 1. Find the distance and midpoint between two points. (A) ( 1, ) and (, ) (B) (, ) and (5, ). (C) (, 4) and (, ) (D) (, 4) and (, 1). Find the center and radius of the following circle. (A) (x 5) + (y + ) = 16 (B) (x + ) + (y ) = 4 (C) x + 16x + y 4y + 16 = 0 (D) x x + y + 8y + 8 = 0 (E) x + 10x + y + 1 = 0 (F) x + y 6y 4 = 0 (G) x 8x + y 6y + 1 = 0 (H) x + y 4x + 8y + = 0 (A) 5, ( 1, 0) (B) 5, ( 7, 0) (C) 9, ( 1, ) (D) 5, ( 1, ) (A) (5, ) r = 4 (B) (,), r = 6 (C) ( 8,), r = 1 (D) (1, 4), r = (E) ( 5,0), r = (F) (0,), r = 1 (G) (4,), r = (H) (1, ), r = 14. Find the equation of circle such that (A) Center: (, 4), Radius: 4 (B) Center: (, ), Contain a point ( 1, ) (C) Center: origin, contain a point (, 4) (D) Center: (, 5), Tangent to y-axis (E) Center: (4, ), Tangent to x = (F) Center: (,4), Tangent to x-axis (G) Two diameter end points at ( 4,) and (, 1) (H) Two diameter end points at (5,1) and ( 1, ) (I) Passes through (4,1), ( 11,6) and (,0) (A) (x ) + (y + 4) = 16 (B) (x ) + (y + ) = 4 (C) x + y = 5 (D) (x ) + (y + 5) = 4 (E) (x 4) + (y + ) = 4 (F) (x + ) + (y 4) = 16 (G) (x + 1) + (y 1) = 1 (H) (x ) + (y + 1) = 1 (I) x + y + x y = 0 4. Find the constant a if the circle x + y + ax 6y = 0 passes through a point (6,) a = 5. Find all points on the x-axis such that the distance between those and a point A(,4) is 5. (1,0), ( 5,0) 6. Find all points with coordinate of the form (a, a) such that the distance between the points (a, a) and (,) is 5. (, ), (,) 7. Find the area of a square which is inscribed in a circle x + y = Find the constant k if x + y + x y + k = 0 and x + y 4x + 6y = 0 touch exteriorly k = 1

56 56 MATH 1111 COLLEGE ALGEBRA REVIEW FOR EXAM 1 Work these specifically. Try them WITHOUT your notes or text. 1. Simplify: (x 5). Factor completely; 6x 19x Factor completely: 4x 6x 4. Find all real/complex solutions: 4x 4x 17 = 0 5. Find all real/complex solutions: x 6x = 4 6. Find all real/complex solutions: x x + 5 = 0 7. Solve the equation: x = x 8. Solve the equation: x 17x + 10x = 0 9. Solve the equation: x x 1x + 8 = Solve the equation: (x 4) (x 4) 4 = Find all real/complex solutions: x 4 5x 6 = 0 1. Solve the equation: x + 6 = x 1. Write 8+5i i in the standard form a + bi 14. Write (6 i)(4 + i) in the standard form a + bi 15. Write i 55 in the standard form a + bi 16. Rationalize the denominator: Simplify the expression: Find the distance and the midpoint between two points (, 5) and (, ) 19. Find the center and radius of the circle x + y + 8x 1y = 0 0. Find the equation of a circle with two diameter end points (6, 4) and (4, 0). 1. Solve the equation: (x + x) x 6x 8 = 0

57 57 SOLUTIONS 1. (x 5)(x 5) = (9x 15x 15x + 5) = 18x 60x x = (x 9) = 4x(x )(x + ). 6x 15x 4x + 10 = x(x 5) (x 5) = (x 5)(x ) 4. x = ( 4)± ( 4) 4(4)( 17) (4) = 4± x = ( 6)± ( 6) 4(1)( 4) (1) = 6± 5 = 6± 1 7. x (x 16) = 0 = ± 1 = 4±1 8 = 1 ± 6. x = ( )± ( ) 4()(5) () = ±6 i 4 = 1 ± i 8. x(x 17x + 10) = 0 = ± 6 4 The solution set is {0, 16} x = 0, x = ( 17)± ( 17) 4()(10) () = 17±1 4 The solution set is {0,, 5} 9. x (x ) 4(x ) = 0 (x )(x 4) = 0 (x )(x )(x + ) = 0 The solution set is {,, } 11. (x ) 5(x ) 6 = 0 (x 6)(x + 1) = 0 x = 6 or x = 1 x ± 6 or x = ± 1 = ±i 10. A A 4 = 0, A = x 4 (A 7)(A + 6) = 0 A = 7, 6 x 4 = 7, x 4 = 6 The solution set is {11, } 1. x + 6 = x, x 0 0 = x x 6 = (x )(x + ), x 0 The solution set is {} The solution set is { 6, 6, i, i} i = (8+5i)(+i) = 16+4i+10i+15i = 1+4i = i ( i)(+i) 4+6i 6i 9i i i 55 = i 5+ = i = i = 5( ) ( ) + 4 = = x + y + 4x 6y = 11 (x + 4x + 4) + (y 6y + 9) = (x + ) + (x ) = 4 (,); r = 4 = (6 i)(4 + i) = i 4i i = i + = i 5 4 = 5(4+ ) = 0+5 = 0+5 (4 )(4+ ) ( ( )) + (( ) 5) = 4 5; ( ( )+, 5+( ) ) = (0,1) 0. Midpoint= center: ( 6+4, ( 4)+0 ) = (5, ) Radius: (5 4) + (( ) 0) = 5 (x 5) + (y + ) = 5 1. (x + x) (x + x) 8 = (x + x 4)(x + x + ) = (x + 4)(x 1)(x + 1)(x + ) = 0 The solution set is { 4,, 1,1}

58 58 MORE PRACTICE PROBLEMS FOR EXAM 1 1. Simplify the expression (A) 5x [x 4y (x y)] (B) 5(x y) 4(x 5y) (C) x y ( xy ) (D) (x y ) 5xy (E) 9xy x y (G) x y 10 xy 5 x y (I) (x 5y) (F) (x y)(5x y) (H) ( x y ) ( y5 x ) (J) (x y)(x + y) (K) ( 5x) (L) ( + x) (A) x + y (B) 6x + 5y (C) x 4 y (D) 5x 5 y 8 (E) y x (F) 10x 19xy + 6y (G) 6xy (H) x 4 y 9 (I) 9x 0xy + 5y (J) x 4 y (K) 5x 0x + (L) 4x + 1x + 9. Simplify and write it in the standard form. (A) i 67 (B) (1 + i) ( 5i) (C) ( i)( i) (D) i 54 (E) 4i i (F) 1 i i (A) i (B) + 7i (C) 4 7i (D) 1 (E) 4 i (F) i. Simplify the following expression. (A) 54 (B) 44 (C) (D) 98 (E) 00 (F) 16 (G) 4 (H) 0.7 (I) (A) 6 (B) 11i (C) 4 i (D) 7 (E) 10 (F) 9 (G) (H) /10 (I) 5/4 4. Rationalize the denominator. (A) 5 7 (B) 4 5 (C) 7 (A) (B) (C) Factor completely. (A) (a b)x (a b)y (B) x + 4x + (C) x 5x + 6 (D) x x 8 (E) x + 11x + 6 (F) x 11x 6 (G) x 6x x + 1 (H) x 9 (A) (a b)(x y) (B) (x + 1)(x + ) (C) (x )(x ) (D) (x 4)(x + ) (E) (x + )(x + ) (F) (x + 1)(x 6) (G) (x 6)(x ) (H) (x )(x + ) 6. Solve each equation. (A) (x 4) + 5 = x 4 (B) (4x ) 1 = 5 (A) x = 1 (B) x = 11 (C) x = ±

59 59 (C) x = 1 (D) (x ) = 4 (E) x 8x = (F) x x = 0 (G) x + 8x = (H) 4x x + = 0 (I) x 7x + 5x = 0 (J) x x 18x + 7 = 0 (K) (x 1) (x 1) 6 = 0 (L) (x + ) + (x + ) 8 = 0 (M) x 4 x 1 = 0 (N) (x x) 8(x x) + 1 = 0 (D) x = ± 6 (E) x = 4± 10 (F) x = 0.5, (G) x = 4± 7 (H) x = 1±i 11 4 (I) x = 0,1, 5 (J) x =,, (K) x = 4, 1 (L) x = 6,0 (M) x = ±, ± (N) x =,,, 1 7. Simplify the expression: (A) (C) (B) (D) (A) 1 (B) 8 (C) 11 5 (D) Solve each equation. (A) x = x (B) 5x 6 = x (C) 8 x = x (D) x 5x + = x (E) x + x + 5 = x + 1 (F) x + x 1 = x (A) x = 1 (B) x =, (C) x = (D) x = 7 (E) x = 4 (F) x = 9. Find the midpoint and distance between ( 4,) and (, 5) ( 1, 1); Find the center and radius of the following circle (A) x + y 4x + 6y = 0 (B) x + y + 6x = 0 (A) (, ); r = 4 (B) ( 1,0); r = 11. Find the equation of a circle which satisfies the following conditions (A) Center (,5) and radius r = 5 (B) Diameter two end points: (,), (4, 1) (C) Diameter two end points: (, 5), (,) (A) (x + ) + (x 5) = 5 (B) (x 1) + (y + 1) = 1 (C) x + (y + 1) = 5 1. Find the equation of a circle such that it satisfies the following 1) its center lies on the graph of y = x + 1, ) it contains a point (,), ) it is tangent to the x-axis. (x 1) + (y ) = 4 or (x 9) + (y 10) = 100

60 60 UNIT : RECTANGULAR SYSTEMS AND FUNCTIONS LECTURE -1 INTERCEPTS, SYMMETRY, AND FUNCTIONS Video 1) Use calculator to find intercepts: ) Finding x, y intercepts ) Decide the symmetry of a graph 4) Even/odd functions QUADRANTS: INTERCEPTS: The x -intercept is where the graph crosses the x axis; x-value when y s value is 0. The y -intercept is where the graph crosses the y axis; y-value when x s value is 0. 4 (0,) : y- intercept x-intercept : (,0) 5 EXAMPLE 1 YOUR TURN 1 Find the intercepts of x 4y = 1 and write your Find the intercepts of x 5y = 0 and write your answer in the ordered pairs. answer in the ordered pairs. x-intercept: when y = 0, it is the value of x. x = 1 x = 4 y-intercept: when x = 0, it is the value of y. 4y = 1 y = Therefore, the intercepts in the ordered pairs are (4,0), (0, )

61 61 EXAMPLE YOUR TURN Find the intercepts of y = in the ordered pairs. x 5 x x 6 and write your answer x-intercept: when y = 0 and x, x, Find the intercepts of the following functions and write your answer in the ordered pairs. (A) y = x 6 x+ it is the value of x. 0 = x 5 (x )(x+) 0 (x )(x + ) = (x 5) (x )(x + ) (x )(x+) 0 = x 5 5 = x (B) y = x+ x +x 1 y-intercept: when x = 0, it is the value of y. y = 5 6 y = 5 6 Therefore, the intercepts in the ordered pairs are (5,0), (0, 5 6 ) EXAMPLE YOUR TURN Find the intercepts of y = x x 8 and write your answer in the ordered pairs. Find the intercepts of y = x x 4 and write your answer in the ordered pairs. x-intercepts: when y = 0, it is the value of x. 0 = x x 8 0 = (x + )(x 4) x =, 4 y-intercept: when x = 0, it is the value of y. y = 8 Therefore, the intercepts in the ordered pairs are (,0), (4,0), (0, 8)

62 6 EXAMPLE 4 YOUR TURN 4 Graph y = x 4 by plotting points 1) Find some points on the graph of y = x 4 x values y values Graph y = x + 1 by plotting points ) Pick those points in the standard (x, y) plane and connect the points smoothly 6 4 y = x SYMMETRY: Symmetry about the x-axis Symmetry about the y-axis Symmetry about the origin Point (a, b) (a, b) ( a, b) ( a, b) Graph y axis y axis y axis (a, b) ( a, b) (a, b) (a, b) x axis (a, b) x axis x axis ( a, b) Whenever (a, b) is on the graph, Whenever (a, b) is on the graph, Whenever (a, b) is on the graph, (a, b) is also on the graph. ( a, b) is also on the graph. ( a, b) is also on the graph. Algebraic Replacing y with y yields an Replacing x with x yields an Replacing y with y and x with x Test equivalent equation. Original : y = x equivalent equation. Original : y = x yields an equivalent equation. Original : y = x Replace y with y: Replace x with x: Replacing y with y and x with x : ( y) = x y = x y = ( x) y = x ( y) = ( x) y = x y = x

63 6 EXAMPLE 5 Decide whether it is the symmetry with respect to x-axis,y-axis, origin, or nether. (A) y axis (B) y axis (C) y axis x axis x axis x axis y axis (D) y axis (E) (F) y axis x axis x axis x axis EXAMPLE 6 YOUR TURN 6 Test graphically the symmetry of the following equation if any: use your calculator (A) y = x + 1 : symmetry w.r.t y-axis Test graphically the symmetry of the following equation if any: use your calculator (A) y = x 1 (B) y = x 6 + 5x 4 y = x (B) y = x + : None (C) y = x 9x (D) y = x + 4 y = x + 4 (C) y = 4x x : symmetry w.r.t origin (E) y = x x +1 5 (F) y = x 4 y = 4 x x 4

64 64 EXAMPLE 7 YOUR TURN 7 Show the expression y = x + x is symmetric with Show the expression y = x 5 is symmetric with respect to respect to the origin. the origin. When x x and y y, it is still same. ( y) = ( x) + ( x) = x x y = x x y = x + x Then it is symmetric with respect to the origin. EXAMPLE 8 YOUR TURN 8 Show the expression y = x + is symmetric with Show the expression y = 5x 4 is symmetric with respect to respect to y-axis. y-axis. When x x, it is still same. y = ( x) + = x + Then, it is symmetric with respect to y-axis. RELATION: A relation is just a set of ordered pairs. FUNCTIONS: A function is a relation between a set of inputs and a set of permissible outputs (codomain) with the property that each input is related to exactly one output. The domain (input) is the set of all input values x to which the rule applies. The range (output) is the set of all output values. FUNCTION NOTATION: The notation y = f(x) is a function whose name is f. This is read as y is a function of x. The letter x represents the input value (independent value) and the letter y or f(x) represents the corresponding output value (dependent value).

65 65 VERTICAL LINE TEST (FUNCTION OR NOT): A graph of a relation is the graph of a function if and only if there is no vertical line that intersects the graph more than one point. FUNCTION FUNCTION NOT A FUNCTION Fail the vertical line test pass the vertical line test pass the vertical line test EXAMPLE 9 YOUR TURN 9 Determine which relation represents y as a function of x Determine which relation represents y as a function of x or not or not (A) (A) y = x (B) x + y = 9 4 (B) (C) By using vertical line test, (A) is a function but (B) is not. EXAMPLE 10 YOUR TURN 10 Determine which relation represents y as a function of x Determine which relation represents y as a function of x or not or not (A) {(,6), (,6), (4,9), (,10)} (A) {( 1,6), (1,5), (,9), (6,)} (B) {(1,), (,), (,), (4,)} (A) Since has two corresponding y-values 6 and 10, it is Not a function. (B) Since every different x values have only one yvalue, it is a function. (B) {(1,), (,), (4,), (1,1)}

66 66 EXAMPLE 11 YOUR TURN 11 Determine which relation represents y as a function of x or not (A) y = x 5 x 4 (C) y = x (B) y = x (D) y = x 4x Determine which relation represents y as a function of x or not (A) y = x (B) y = x 1 x+ After we solve for y and we can express y as only one expression of x, we can say it is a function. (A) It is a function (B) Since y = ± x, it is Not a function. (C) Since y = ±x, it is Not a function. (D) It is a function. (C) y = x (D) x + 4y = 6 (E) y = x + (F) y = 4 x

67 67 PRACTICE PROBLEMS 1. Find the intercepts (A) y = x x 4 (C) y = x 4 8x 9 (B) y = x 4 x+8 (D) y = x x 1 x+5 (E) y = x 1 (F) y = x + 8 (G) 5x 4y = 8 (H) y = x + 6 (I) x + y = 4 (J) x + y = 16 (K) x + 9y = 9 (L) y = x x +4x 1 (A) x int: 4, 1; y int: 4 (B) x int: 4; y int: 1 (C) x int:,, y int: 9 (D) x int: 4,, y int: 1 (E) x int: 1, y int: 1 (F) x int:, y int: 8 (G) x int: 8, y int: 5 (H) x int:, y int: 6 (I) x int:, y int: 4 (J) x int: ±4 y int: ± 4 (K) x int: ± y int: ± 1 (L) x int:, y int: Find the symmetry if any (A) 5 (B) (C) y = x + (D) y = x x +1 (E) y = x (F) y = x (A) x axis (B) None (C) y axis (D) origin (E) origin (F) y axis (G) origin (H) y-axis (I) x-axis (J) None (K) None (L) x axis, y axis, origin (G) y = 9x x (H) y = x + 1 (I) x y = 4 (J) y = x + 4 (K) y = x (L) x + y = 16. Decide whether the relation represents y as a function of x. (A) (B) (A) (B) (C) (D) (E) (F) FUN NOT A FUN NOT A FUN FUN NOT A FUN FUN

68 68 (C) (D) (E) (F) 4. Decide whether the relation represents y as a function of x. (A) {(,), (, 5), (4,), (5, )} (C) y = x (D) x = y (E) y = x 4 x+8 (B) {(, 4), (,), (,5), (4, )} (F) y = x x + 6 (G) x + y = 5 (H) y = 5x (I) y = x + (J) x 4y = 8 (K) 4x 5y = 0 (L) y = x x+5 (M) y = 10 x+4 (N) y = x 5 5. Decide whether the following function is even/odd/neither. Then describe the symmetry (A) y = x 5 (B) y = x 4 (C) y = x + (D) y = x + (E) y = x 4x (F) y = x (G) y = x +x x + (I) y = x +1 (H) x + y = 5 (J) y = x 9x 6. Decide whether the following function is even/odd/neither algebraically (A) y = x x +4 (C) y = x x (B) y = x 4 + x (D) y = x + x (A) FUN (B) NOT A FUN (C) FUN (D) NOT A FUN (E) FUN (F) FUN (G) NOT A FUN (H) FUN (I) FUN (J) FUN (K) NOT A FUN (L) FUN (M) FUN (N) NOT A FUN (A) Odd (B) Neither (C) Even (D) Neither (E) Odd (F) Even (G) Odd (H) Not function (I) Even (J) Neither (A) f( x) = f(x); odd (B) f( x) = f(x): even (C) f( x) ±f(x): Neither (D) f( x) = f(x); odd

69 69 LECTURE - DOMAIN AND RANGE OF FUNCTIONS Video 1) Find the domain of a function INTERVAL NOTATION: NOTATION INEQUALITY GRAPH (a, b) a < x < b a b (a, b] a < x b a b [a, b) a x < b a b [a, b] a x b a b (a, ) a < x a b [a, ) a x a (, a) x < a a (, a] x a a PROPERTIES OF INEQUALITY (A) Transitivity: a < b, b < c a < c (B) Operations ADDITION SUBTRACTION a < b a + c < b + c a c < b c MULTIPLICATION DIVISION ac < bc, c > 0 { ac > bc, c < 0 a c { < b, c > 0 c a > b, c < 0 c c

70 70 EXAMPLE 1 YOUR TURN 1 Solve the inequality and write your solution by using Solve the inequality and write your solution by using interval notations. interval notations. (5x + 1) + 1 (7x ) (A) 0 6x (5 + 7x) (5x + 1) + 1 (7x ) 15x x 9 Distribute 15x x 9 6x x 5 x 5 6 Subtract 1x from each side Subtract 16 from each side Divide each side by 6 (negative) Reverse the inequality (B) x > 1 4 The solution set is the interval [ 5 6, ).

71 71 NOTE: The domain (input) is the set of all input values x to which the rule applies. The range (output) is the set of all output values HOW TO FIND DOMAIN OF A FUNCTION: Case Domain Polynomial case All real numbers Fraction case All real numbers except the x-values which make denominator zero; Square root case All real numbers which make the inside of square root non-negative; EXAMPLE YOUR TURN Find the domain and range of a function Find the domain and range of a function f = {( 1,), (,), (4, 1), (6, )} f = {(,5), ( 1,5), (, ), (4, )} Since domain is the collection of all different x values, Domain of f = { 1,,4,6} Since range is the collection of all different y values, Range of f = { 1,,} EXAMPLE YOUR TURN Find the domain and range of a function in the given Find the domain and range of a function in the given figure. figure Domain: (0, ) and Range: (, )

72 7 EXAMPLE 4 YOUR TURN 4 Find the domain and range of a function in the given figure. Find the domain and range of a function in the given figure. Domain: the collection of all x-values (from left to right) [ 4, 6] Range: the collection of all y-values (from down to up) [,] FORMULA CASE EXAMPLE 5 YOUR TURN 5 Find the domain of a function y = 5x 5 4x + 6. Find the domain of a function y = 4x Since it is a polynomial, its domain is (, ). EXAMPLE 5 YOUR TURN 5 Find the domain of a function y = x+6 x x 1 1) Find all x-values which is not in domain. Find the domain of a function (A) y = x x 6 denominator 0 x x 1 0 (x 4)(x + ) 0 x 4, (B) y = x+5 x x 4 ) Domain is all real numbers except 4, ; (, ) (,4) (4, )

73 7 EXAMPLE 7 YOUR TURN 7 Find the domain of a function y = x 8 Find the domain of a function y = x 1 Since it is a radical expression, x 8 0 x 8 x 4 So, its domain is [4, ) EXAMPLE 8 YOUR TURN 8 Find the domain of a function y = 1 4x Find the domain of a function y = 10 5x Since it is a radical expression, 1 4x 0 4x 1 x 1 4 = So, its domain is (, ]. Reverse inequality by dividing 4 EXAMPLE 9 YOUR TURN 9 Find the domain of a function y = x Fraction condition x Square root condition x 0 x 0 x 0 Find the domain of a function y = x+6 x It satisfies both conditions: x 0 and x 0 x > 0. Therefore, its domain is (0, ) EXAMPLE 10 YOUR TURN 10 Find the domain of a function y = x+ x Find the domain of a function y = x 1 x 5 Fraction condition x 0 x Square root condition x + 0 x It satisfies both conditions that x and x. Therefore, its domain is [,) (, ).

74 74 THE GRAPH OF A FUNCTION: f(a) = b means a point (a, b) which is on the graph of f The graph of a function f whose domain is X means the collection of all points (x, f(x)), x X in the plane EXAMPLE 11 YOUR TURN 11 Decide whether the following statement is true or false. Decide whether the following statement is true or false. (A) A point (,4) is on the graph of f(x) = x 1. (A) A point ( 1,5) is on the graph of f(x) = x +. (B) f(x) = x x + 5 contains a point ( 1,10). (A) If (,4) is on the graph of y = f(x), it means f() = 4. But f() = () 1 = 5 4 It is a false statement. (B) f(x) = 5x x + contains a point (,5). (B) If ( 1,10) is on the graph of y = f(x), It means f( 1) = 10. f( 1) = ( 1) ( 1) + 5 = 10 It is a true statement. EXAMPLE 1 YOUR TURN 1 If the graph of the function If the graph of the function f(x) = a(x ) (x + 1) + 1 f(x) = a(x ) (x + 1) + 6 contains the point (0, 9), then find the constant a. contains the point (,4), then find the constant a. Since (,4) is on the graph of f, f() = 4 a( ) ( + 1) + 6 = 4 a ( 1) () + 6 = 4 = a + 6 = 4 a = 18 a = 6

75 75 EXAMPLE 1 YOUR TURN 1 If f(x) = kx + x kx +, find a number k such that If f(x) = x kx + x 5k, find a number k such that the graph of f contains a point (,1). the graph of f contains a point ( 1,4). Since f contains a point (,1), f() = 1 k() + () k() + = 1 6k + 6 = 1 6k = 6 k = 1 EXAMPLE 14 YOUR TURN 14 If one zero of f(x) = x x 16x + 16k is, find If one zero of f(x) = x x kx + 1 is, find two other zeros. two other zeros. Since f contains a point (,0), f() = 0 () () 16() + 16k = 0 16k = 0 k = Then f(x) = x x 16x + x x 16x + = x (x ) 16(x ) Grouping = (x )(x 16) A B = (x )(x 4)(x + 4) Therefore, the other two zeros are 4, 4

76 76 PRACTICE PROBLEMS 1. Find the domain of the following function. (A) {(,), (, 5), (1,), (4, )} (B) {(1,), (,4), (4,5), (5,4)} (C) f(x) = 5x 7x + 8 (D) f(x) = x + 6 (E) f(x) = 16 x (F) f(x) = 5x (G) f(x) = x x+6 (I) f(x) = x x 8 (H) f(x) = x+ x +x 1 (J) f(x) = x 4 x+1 (K) f(x) = x 9 (L) f(x) = x+ x 5 (M) f(x) = x 7 x, x < 0 (O) f(x) = { x, x 0 (N) f(x) = 8x x +16 x 1, < x 1 (P) f(x) = { x +, 1 < x (A) {1,,,4} (B) {1,,4,5} (C) (, ) (D) [, ) (E) (, 8] (F) [, ) 5 (G) (, 6) ( 6, ) (H) (, 4) ( 4,) (, ) (I) [0,8) (8, ) (J) ( 1, ) (K) (, ] [, ) (L) (, 5) ( 5,5) (5, ) (M) (7, ) (N) (, ) (O) (, ) (P) (,]. Find the domain of the graph of the function. (A) (B) (C) (D) (A) Domain: [,] Range:[,] (B) Domain: (, ) Range:[, ) (C) Domain: (0, ) Range:(, ) (D) Domain: [ 1, ) Range:[0, ) (E) Domain: (,] Range:[ 1,] (F) Domain: (, ) Range: (, ). Find the constant k if a point (,) is on the graph of f(x) = x kx + 4 k = 7 4. Find the constant k if a point (,1) is on the graph of f(x) = x k x+ k = 5. Find the constant k if a point (,1) is on the graph of f(x) = x + kx + x 7 k = Find values of k such that f(x) = x + kx k x 6 and f( ) = 0 k = 5, 7. Find values of k such that f(x) = 5x + k x 8kx 8 and f( ) = 0 k = 6,

77 77 LECTURE - NOTATION OF FUNCTIONS AND PIECEWISE FUNCTIONS Video 1) Function notation ) Piecewise functions EXAMPLE 1 YOUR TURN 1 Consider f(x) = x + x 4. Find Consider f(x) = x 5x +. Find (A) f( ) (B) f(x) (A) f( ) (C) f( x) (D) f(x ) (A) f( ) = ( ) + ( ) 4 (B) f(x) = = 17 (B) f(x) (C) f(x ) = (x) + (x) 4 = (x)(x) + 4x 4 = 1x + 4x 4 (C) f( x) = ( x) + ( x) 4 = x x 4 (D) f(x ) (D) f( x) = (x ) + (x ) 4 = (x )(x ) + (x ) 4 = (x 4x + 4) + x 4 4 = x 1x x 8 = x 10x + 4

78 78 EXAMPLE YOUR TURN Consider f(x) = 5x. Find (A) f( ) (B) f( x ) Consider f(x) = x + 7. Find (A) f( ) (C) f(x) (D) f(x) (A) f( ) = 5( ) = 17 (B) f( x ) (B) f( x) = 5( x) = 5x (C) f(x) = 5(x) = 5(x)(x) (C) f(5x) = 0x (D) f(x) = {5x } = 15x 9 (D) f(x) EXAMPLE YOUR TURN Find f(x+h) f(x) h if f(x) = 4x + Find f(x+h) f(x) h if f(x) = x 5 1) Find f(x + h) f(x + h) = 4(x + h) + = 4x + 4h + ) Find f(x+h) f(x) h f(x + h) f(x) [4x + 4h + ] [4x + ] = h h 4x + 4h + 4x = h = 4h h = 4

79 79 EXAMPLE 4 YOUR TURN 4 x Find f() if f(x) = {, x > 1 x +, x 1. x Find f() if f(x) = {, x < 1 x +, x 1. Since > 1, we use f(x) = x ; f() = () = 8 EXAMPLE 5 YOUR TURN 5 x, x < 0 Find f() if f(x) = { 8, x = 0 x + 1, x > 0 x, x < 0 Find f( ) if f(x) = { 8, x = 0 x + 1, x > 0 Since > 0, we use f(x) = x + 1; f() = () + 1 = 7 EXAMPLE 6 YOUR TURN 6 x, x < Find f( ) if f(x) = { 5, x = x, x > x, x > 1 Find f(1) if f(x) = { 5, x = 1 x, x < 1 Since <, we use f(x) = x ; f( ) = ( ) = 7 YOUR TURN x, x < Find f(4) if f(x) = { 4, x < x 7, x YOUR TURN Find f( 1) if f(x) = { x, x < 4, x < x 7, x

80 80 EXAMPLE 7 YOUR TURN 7 Sprint offers a monthly cellular phone plan for $9.99. It Sprint offers a monthly cellular phone plan for $9.99. It includes 450 minutes and charges $0.45 per minute for includes 450 minutes and charges $0.45 per minute for additional minutes. The following is used to compute the additional minutes. The following is used to compute the monthly cost for subscriber monthly cost for subscriber 9.99, 0 x , 0 x 450 C(x) = { C(x) = { 0.45x 16.51, x > x 16.51, x > 450 where x is the number of minutes used. Compute the where x is the number of minutes used. monthly cost of the cellular phone for 480 minutes (A) Compute the monthly cost of the cellular phone for 80 minutes Since 480 > 450, C(x) = 0.45x C(480) = 0.45(480) = 5.49 (B) Compute the monthly cost of the cellular phone for 500 minutes EXAMPLE 8 YOUR TURN 8 1, x < 1 Draw the graph of f(x) = { x 1, x 1 Draw the graph of f(x) = { x 1, x < 0 x, x 0 When x < 1, the function is y = 1. x y 1 1 Open point (x < 1) 1 1 When x 1, the function is y = 1. x y closed point (x 1)

81 81 ZERO(S) OF A FUNCTION: Zeros of a function f are the x value(s) such that f(x) = 0 A point (a, b) is on the graph of y = f(x) if b = f(a) EXAMPLE 9 YOUR TURN 9 Consider f(x) = x 4x. Consider f(x) = 4x + x. (A) Decide whether a point ( 1,) is on the graph of f? (A) Decide whether a point (8,0) is on the graph of f? (B) Find f( ) (C) If f(x) = 5, what is x? (D) Find x-intercepts. (E) Find the zero(s) of f. (F) Find the domain of f (B) Find f( ) (A) Since f(1) = ( 1) 4( 1) =, a point ( 1,) is NOT on the graph of f. (B) f( ) = ( ) 4( ) = = 1 (C) If f(x) = 6, what is x? (C) x 4x = 5 x 4x 5 = (x 5)(x + 1) = 0 Then, x = 5, 1 (D) Find x-intercepts. (D) x-intercepts are x values when y = 0. x 4x = x(x 4) = 0 x = 4, 0 (E) Since x-intercepts are zeros, x = 4, 0 (F) Since it is a polynomial, its domain is (, ) (E) Find the zero(s) of f. (F) Find the domain of f

82 8 EXAMPLE 10 YOUR TURN 10 Consider f(x) = x+ x 6. (A) Decide whether a point (4, ) is on the graph of f? (B) Find f(5) (C) If f(x) =, what is x? (D) Find x-intercepts. (E) Find the zero(s) of f. (F) Find the domain of f Consider f(x) = x 8 x+. (A) Decide whether a point (,1) is on the graph of f? (B) Find f() (A) Since f(4) = 4+ = 6 =, it is on the graph. 4 6 (B) f(5) = = 7 1 = 7 (C) If f(x) =, what is x? (C) f(x) = x + x 6 = 1 x + = (x 6) x + = x 1 14 = x (D) Find x-intercepts. (D) x-intercepts are x values when y = 0 x + x 6 = 0 1 (since 0 = 0 1 ) x + = 0 (x 6) x + = 0 x = (E) Find the zero(s) of f. (E) Since x-intercepts are zeros, x = (F) Find the domain of f (F) Since it is a rational function, x 6 Its domain is (, 6) (6, )

83 8 PRACITCE PROBLEMS 1. Let f = {(,), ( 1,5), (0,4), (1, ), (,5)}. Find the following. (A) f( ) (B) f(0) (C) f(1) (D) f(). Let f(x) = x+. Find the following x 1 (A) f() (B) f( ) (C) f(1) (D) f(). Let f(x) = x 4x + 5. Find the following (A) f() (B) f( ) (C) f(4x) (D) f( x) 4. Let f(x) = x 6x 5. Find the following. (A) f( 4) (B) f(x) (C) f(x ) (D) f(x) 5. Let f(x) = 5x + x 7. Find the following. (A) f() (B) f( 5) (C) f(5x) (D) f(x + ) 6. Let f(x) = x 7. Find the following. (A) f( ) (B) f( x) (C) f( x ) (D) f(x 1) 7. Find f(x+) f() x if f(x) = x x + 5 (A) (B) 4 (C) (D) 5 (A) (B) 0 (C) Undefined (D) 5 (A) 11 (B) 5 (C) x 16x + 5 (D) x + 4x + 5 (A) 67 (B) 1x 1x 5 (C) x 18x + 19 (D) 6x + 1x + 10 (A) 44 (B) 108 (C) 15x + 10x 7 (D) 5x + x + 17 (A) 5 (B) 1x 7 (C) x 1 (D) 9x 18x x x 8. Let f(x) =, x < 1 { x +, x 1. Evaluate (A) f(1) (B) f( ) (C) f( 1) (D) f(6) 1 9. Let f(x) = {, x 1 x 1 5, x = 1. Evaluate (A) f(1) (B) f() (C) f ( 1 ) (D) f() x +, x < Let f(x) = { 8, 1 < x <. Evaluate x, x (A) f(4) (B) f() (C) f(1) (D) f( ) x 1, if x < Let f(x) = {, if x = 0 x 1, if x > 0 Evalute (A) f( 1) (B) f() (C) f(0) (D) f() (A) f(1) = (B) f( ) = 8 (C) f( 1) = 1 (D) f(6) = (A) f(1) = 5, (B) f() = 1 (C) f ( 1 ) = (D) f() = 1 (A) f(4) = 1 (B) f() = 0 (C) f(1) = 8 (D) f( ) = 6 (A) f( 1) = 0 (B) f() = 8 (C) f(0) = (D) f() = 5

84 84 1. Let f(x) = x + 5x 9 and g(x) = 4x +. Solve (A) g(x) = 0 (B) f(x) = g(x) (A) 4 (B) 4, 1. Find the real zero(s) of the function. (A) y = x 4x + (C) y = x x+4 (B) y = x 16x (D) y = x x 6 x 9 (A) ± (B) 0, 4,4 (C) (D) 14. Find the intersection point(s) of y = x x and y = x (4,8), (,10) 15. Using the graph of y = f(x), (A) Find f( 1) (B) Find f() (C) Find x-value such that f(x) = 4 (D) Find x-value(s) such that f(x) = 0 (E) Find x-intercept(s) 4 y = f(x) (A) 1 (B) (C) (D),0,4 (E),0,4 16. Consider the function f(x) = x x. (A) Is the point (1, ) on the graph of f? (B) If x =, what is f(x)? (C) If f(x) = 5, what is x? (D) Find the zeros (x-intercepts) of f. (E) Find the domain of f. (A) Not (B) 1 (C),4 (D), 1 (E) (, ) 17. Consider the function f(x) = x x 4. (A) Is the point (1, 1 ) on the graph of f? (B) Find the value of f(x) when x =. (C) If f(x) = 1, what is x? (D) Find the zeros (x-intercepts) of f if any. (E) Find the domain of f. (A) Yes (B) undefined (C) 1 (D) None (E) (, ) (,) (, ) 18. To solve the equation f(x) = g(x), use the graphs of functions in the figure. (A) y = f(x) (B) (A) x = 1, (B) x =, y= g(x) y = g(x) 4 y = f(x)

85 85 LECTURE -4 OPERATIONS OF FUCTIONS AND PROPERTIES OF FUNCTIONS Video 1) Function operations ) Increasing/decreasing/constant and local max/min OPERATION OF FUNCTIONS: Let f and g be functions. (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) ( f f(x) ) (x) =, g(x) 0 g g(x) The domain of the sum, difference, or product of functions is intersection of domain of f and domain of g. The domain of the division of functions is intersection of domain of f and domain of g except x such that g(x) 0. EXAMPLE 1 YOUR TURN 1 Let f(x) = x and g(x) = x + 1. Find Let f(x) = x 1 and g(x) = x + 5. Find (A) (f + g)(x) (A) (f + g)(x) = f(x) + g(x) = (x ) + (x + 1) = 5x Domain of f + g is (, ) (B) (f g)(x) = f(x) g(x) = (x ) (x + 1) = x x 1 = x 4 Domain of f g is (, ) (C) (f g)(x) = f(x) g(x) = (x ) (x + 1) = 6x + x 9x = 6x 7x Domain of fg is (, ) (D) ( f f(x) ) (x) = = x g g(x) x+1 Since the domain satisfies the denominator 0, (B) (f g)(x) (C) (f g)(x) (D) ( f g ) (x) Domain of f g is (, 1 ) ( 1, )

86 86 EXAMPLE YOUR TURN Let f(x) = x and g(x) = x + 1. Find (A) (f g)() (B) (f g)() Let f(x) = x 1 and g(x) = x + 5. Find (A) (f g)() (A) (f g)() = f() g() = (4 ) (6 + 1) = 1 7 = 6 (B) (f g)() = f() g() = (6 ) (9 + 1) = 10 = 0 (B) (f g)() EXAMPLE YOUR TURN Find the domain of fg if f(x) = x and g(x) = 1 x 5 Find the domain of fg if f(x) = 1 and g(x) = x x+5 Since (fg)(x) = f(x) g(x) = x is a fraction, x 5 Its domain satisfies x 5 0 x 5 x 5 Therefore, its domain is (, 5 ) (5, )

87 87 EXAMPLE 4 YOUR TURN 4 Find the domain of f g if Find the domain of f g if f(x) = x and g(x) = x 4 x+ f(x) = 5x 1 and g(x) = x+1 x We know that ( f g ) (x) = x x 4 x+ We know that a fraction is defined at all values which denominator 0; We consider two things; 1) ) x ( x 4 x+ ) x 4 x+ x 4 is a fraction 0 x 4 0 x+ is a fraction x + 0. Therefore, x, x 4 The domain of f is (, ) (,4) (4, ) g EXAMPLE 5 YOUR TURN 5 Find the domain of f g if Find the domain of f g if f(x) = x+ x 4 and g(x) = x+6 x+ f(x) = x 5 x+1 and g(x) = 1 x 7 x We know that ( f g ) (x) = x+ x+6 ( x 4 x+ ) We know that a fraction is defined at all values which denominator 0. We consider three things; 1) ) x+ x+6 ( x 4 x+ x 4 is a fraction 0 ) x+ x 4 x + 4 x 4 x 8 x+ x 4 x+ ) x+ x+6 is a fraction x + 0. is a fraction x Therefore, x 6, x, x 8 The domain of f g is (, 6) ( 6, ) (,8) (8, )

88 88 INCREASING/ DECREASING/ CONSTANT 1) A function f is increasing on an interval if for any a and b in this interval, a < b implies f(a) < f(b) ) A function f is decreasing on an interval if for any a and b in this interval, a < b implies f(a) > f(b) ) A function f is constant on an interval if for any a and b in this interval, a < b implies f(a) = f(b) EXAMPLE 6 YOUR TURN 6 Find the increasing/decreasing/constant intervals of the Find the increasing/decreasing/constant intervals of the graph of a function. graph of a function ) Find the turning points (local max/min) 4 y y = f(x) increasing increasing 5 5 decreasing decreasing x 4 ) This is the intervals of x-values Increasing intervals:(,0) (, ) Decreasing intervals: (, ) (0,) No constant intervals

89 89 LOCAL MAX/MIN (TURNING POINTS) 1) A function f has a local maximum at c if for any x in this interval, f(x) < f(c) ) A function f has a local maximum at c if for any x in this interval, f(x) > f(c) ABSOLUTE MAX/MIN 1) A function f has an absolute maximum at c if for any x in the domain, f(x) < f(c) ) A function f has an absolute maximum at c if for any x in the domain, f(x) > f(c) EXAMPLE 7 YOUR TURN 7 Find the local maximum and local minimum of a function Find the local maximum/minimum and the absolute in the figure if any maximum/minimum of a function in the figure if any We can find two turning points (local max/min) The graph has a local maximum 4 at x = since it is the highest point in the open interval around x =. The graph has a local minimum at x = since it is the lowest point in the open interval around x = 1. But the points (0,1), (5,5) are not local max/min since it does not have left or right-side graph. Since the graph has a highest point (5,5), the absolute maximum is 5 at x = 5. Since the graph has a lowest point (0,1), the absolute minimum is 1 at x = 0.

90 90 Example 8: Use the graph of a function y = f(x) to answer the following questions (A) Find f( ) (B) Find f(5) (C) Find the intercepts of f. (D) Find the domain of f. (E) Find the range of f. (F) For what values of x is f(x) > 0? (G) How often does the line y = 1 (horizontal line) intersect the graph? (H) For what values of x is f(x) =? (I) Find local maximum/minimum (J) Find absolute maximum/minimum

91 91 PRACTICE PROBLEMS 1. Let f(x) = x 5 and g(x) = x 7. Find the following (A) (f + g)(x) (B) (f g)(x) (C) (f g)(x) (D) ( f ) (x) g (E) (f + g)() (F) (f g)( ) (A) 5x 1 (B) x + (C) 6x 1x + 5 (D) (E) x 5 x 7 (F) 0 (G) 4 (H) 7 (G) (f g)() (H) ( f g ) (4). Let f(x) = x and g(x) = 4 x. Find the following (A) (f + g)(x) (B) (f g)(x) (C) (f g)(x) (D) ( f ) (x) g (E) (f + g)() (F) (f g)( ) (A) x + 1 (B) x 7 (C) x + 11x 1 (D) x 4 x (E) (F) 1 (G) (H) undefined (G) (f g)() (H) ( f g ) (4). Let f(x) = 1 x 1 (A) f + g (C) fg and g(x) = x. Find the domain of (B) f g (D) f g (A) (, 1) (1, ) (B) (, 1) (1, ) (C) (, 1) (1, ) (D) (, 1) (1,) (, ) 4. Using the graphs of functions f and g, (A) Find (f + g)() (B) Find (f g)( ) y = f(x) (A) (B) 4 (C) 1 (D) 1 (C) Find (fg)(0) 5 (E) (D) Find ( f g ) (6) y = g(x) 5. Find the increasing/decreasing/constant intervals of the graph of the function. (A) 4 5 (B) 4 (A) IN: (0,) DI: (,5) CI: (5,7) (B) IN: (0,) (5, ) DI: (,) CI: (,5) 5

92 9 6. Find the local maximum and local minimum of a function in the figure if any (A) 4 (B) 4 (A) Local max 4 at x = 1 and Local min 0 at x = 1 (B) Local max at x = 1 and Local min 1 at x = Find absolute maximum /minimum of a function in the graph, if any. (A) 4 (B) 4 (A) (B) Absolute min 1 at x = and No absolute max. Absolute max 4 at x = and No absolute min. 4 4

93 9 LECTURE -5 TRANSFORMATIONS TRANSFORMATION POINTS FORMULA RELATION x place y place Vertically c units (x, y) (x, y + c) x y c SHIFT Horizontally c units (x, y) (x + c, y) x c y x-axis (x, y) (x, y) x y REFLECTION y-axis (x, y) ( x, y) x y Vertically 1 c factor (x, y) (x, y c ) x cy COMPRESS (c > 1) Horizontally 1 c factor (x, y) ( x c, y) cx y STRETCH (c > 1) Vertically c factor (x, y) (x, cy) x Horizontally c factor (x, y) (cx, y) x c y c y

94 94 Video 1) Shift transformations ) Reflection ) Stretch/compress EXAMPLE 1 YOUR TURN 1 Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x 5, but is shifted up 5 units. but is shifted down 4 units. Since y value increases 5, y = ( x + 4) + 5 = x + 9 EXAMPLE YOUR TURN Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x, but is stretched vertically by factors. but is stretched vertically by factors. Since y value becomes twice bigger, y = ( x + 4) = x + 8 EXAMPLE YOUR TURN Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x 5, but is shifted left 5 units. but is shifted right 4 units. Since this is horizontal transformation, replace x by x + 4 y = x EXAMPLE 4 YOUR TURN 4 Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x, but is stretched horizontally by factors. but is stretched horizontally by factors. Since this is horizontal transformation, replace x by x y = x + 4

95 95 EXAMPLE 5 YOUR TURN 5 Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x, but is reflected about the x-axis. but is reflected about the x-axis. Since this is reflection, replace y by y y = ( x + 4) y = x 4 EXAMPLE 6 YOUR TURN 6 Write a function whose graph is the graph of y = x + 4, Write a function whose graph is the graph of y = x, but is reflected about the y-axis. but is reflected about the y-axis. Since this is reflection, replace x by x y = x + 4 EXAMPLE 7 YOUR TURN 7 The graph of y = x is shifted left units, shifted up by The graph of y = x is shifted left by 4 units, reflected units, reflected about the y-axis. Write the resulting across the x-axis, and shifted up by 5 units. Write the equation. resulting equation. The resulting function left units x x + y = x + up units y increases y = x + + Reflect y axis x x y = x + + The resulting function is y = x + +.

96 96 EXAMPLE 8 YOUR TURN 8 The graph of y = x is reflected about the y-axis, shifted The graph of y = x is reflected across by x-axis, shifted left units, shifted up by units. Write the resulting right by units, shifted up 1 unit. Write the result equation. equation. The resulting function Reflect y axis x x y = x left units x x + y = (x + ) up units y increases y = x + Then the resulting function is y = x + EXAMPLE 9 YOUR TURN 9 The graph of y = x is shifted right 5 units, shifted down The graph of y = x is shifted right by units, reflected by 4 units, stretched vertically by a factor of. Write the across the x-axis, and shifted up 5 units. Write the resulting equation. resulting equation. The resulting function right 5 units x x 5 y = (x 5) down 4 units y decreases 4 y = (x 5) 4 stretched vertically by factor times of y y = {(x 5) 4} y = (x 5) 1 Then the resulting function is y = (x 5) + 1

97 97 YOUR TURN The graph of y = x is shifted left by units, reflected across the x-axis, and shifted down 4 units. Write the resulting equation. YOUR TURN The graph of y = x is shifted up by 5 units, stretched vertically by factor 4, and shifted left units. Write the resulting equation. EXAMPLE 10 YOUR TURN 10 The graph of y = (x 4) + 5 can be obtained by the The graph of y = (x + ) 7 can be obtained by the transformations of g(x) = x. What kind of the transformations of g(x) = x. What kind of transformations must be used? transformations must be used? The order of transformation Shift right or left y = (x 4) + 5 Stretch/compress y = (x 4) + 5 Shift up/down y = (x 4) + 5 Transformation; First, shift right 4 units Second, stretch vertically by factors Then, shift up 5 units

98 98 PRACTICE PROBLEMS 1. Let P(,4) be a point of y = f(x). Find the corresponding point on the graph of the given function. (A) y = f(x ) (B) y = f(x + ) (C) y = f(x) 4 (D) y = f(x) + (E) y = f(4x) (F) y = f(x) (G) y = f( x) (H) y = f(x) (A) (4,4) (B) ( 1,4) (C) (,0) (D) (,6) (E) ( 1, 4) (F) (,1) (G) (,4) (H) (, 4) (I) (,5) (J) (6,1) (I) y = f(x) (J) y = f(x 4) + 1. Find the resulting equations after we apply the following transformation to y = x (A) shifted right by 4 units, reflected across the y-axis, and shifted down by units. (B) shifted down by units, stretched vertically by factor, and shifted left by 4 units. (C) shifted left by 4 units, shifted up by units, and stretched vertically by a factor of 5. (D) shifted left by units, shifted up by 5 units, and reflected across the x-axis. (E) shifted up by units, stretched horizontally by a factor of, and shifted right by 4 units (F) shifted left by units, compressed horizontally by a factor of 1, and shifted down 4 units. (A) y = ( x 4) (B) y = (x + 4) 6 (C) y = 5(x + 4) + 15 (D) y = (x + ) 5 (E) y = ( 1 x ) + (F) y = (x + ) 4 (G) y = (x 4) (H) y = ( x ) + 5 (I) y = (x + 7) (J) y = (x+11) (G) shifted up by units, reflected across the x-axis, and shifted right by 4 units. (H) shifted right by units, reflected across the y-axis, and shifted up by 5 units (I) shifted left by 7 units, reflect across the x-axis, and shifted down by units (J) compressed vertically by a factor of 1, reflected across the x-axis, and shifted left by 11 units.. Write the equation of the following functions, given the original function and the transformations performed. (A) f(x) = x; shift left by 6 units, and shift down by 4 units. (B) f(x) = x ; reflect across the x-axis, stretch vertically by a factor of, shift left by 1 unit and up by units (A) y = x (B) y = x (C) y = 1 5x 10 1 (D) y = x + 4 (C) f(x) = 1 ; reflect across the y-axis, compressed horizontally by a factor of 1, shift right x 5 by units and down by 1 unit (D) f(x) = x; shift left units, reflect across the x-axis, and shift down 4 units

99 99 4. Describe the transformations that have been applied to obtain the function from the given base function. (A) y = (x ) ; y = x (B) y = (x + 4) 5; y = x (C) y = 4 x + ; y = x (D) y = 1 (x + ) 4; y = x 5. Describe the transformations that have been applied to obtain the function from y = f(x). (A) y = f(x + ) (B) y = 4f(x 5) (C) y = f(x ) + (D) y = f(x) 4 (A) Shift right units, then vertically stretch by a factor of (B) Shift left 4 units, then down 5 units (C) Shift right units, vertically stretch by a factor of 4, and then shift up units (D) Shift left units, vertically compressed by a factor of 1, and then shift down 4 units. (A) Shift left units, then down units (B) Shift right 5 units, then vertically stretch by a factor of 4 (C) Shift right units, reflect about x-axis, then up units (D) Horizontally compressed by a factor of 1 and shift down 4 units 6. Use the graph of f to draw each graph. 6 (A) y = f(x) + 5 C A (B) y = f(x ) (C) y = f(x) (D) y = f(x) 5 10 E D B (E) y = f ( 1 x) 7. Use the graph of f to draw each graph y = f(x) (A) y = f(x + ) (B) y = f(x) (C) y = f(x) A B 5 5 C E 5 (D) y = f ( 1 x) D 4 (E) y = f(x)

100 100 MATH 1111 COLLEGE ALGEBRA REVIEW FOR EXAM Work these specifically. Try them WITHOUT your notes or text. 1. Find the intercepts of y = x+6 x. If the graph of the function y = a(x + )(x ) contains (passes through) the point (1,18), then find the constant a.. Decide the following is a function or not (A) (B) 4. Which of the following is a function? (A) y = x + (B) y = x 4 (C) y = x 4 x 7 (D) None of above are functions. 5. Evaluate f( ) if f(x) = x 5x Find f(x) if f(x) = x x Evaluate f( ) if f(x) = { 7 x if x < 1 4 if 1 x < x 1 if x 8. Find f(x ) if f(x) = x x Let f(x) = 5 x and g(x) = x +. Then find (A) (f g)() (B) (f g)(x) 10. Given the graph of the function in the figure, which of the following is true? (A) Its domain is [,] (B) Its range is [ 4,6] (C) It is only decreasing on the interval (,) (D) It is only increasing on the interval (,) (E) The local max is at x = 5

101 Find the domain of f(x) = 15 x 1. Find the domain of f(x) = x+5 x x 8 1. Find the domain of f(x) = x x+ 14. Find all function which is symmetric with respect to the origin (an odd function). (A) y = x + (B) y = x + (C) y = x + x (D) y = x 4x (E) y = 1 x (F) y = x x Show that f(x) = x 4 + x is an even function algebraically 16. The graph of y = x is shifted left by units, reflected across the x axis, and shifted down 5 units. Write the resulting equation. 17. The graph of y = x is reflected across the x-axis, shifted right by 4 units, and shifted up 5 units. Write the resulting equation. 18. The graph of the function y = (x + 4) 4 + can be obtained from the graph of the function y = x 4 by which of the following transformations? (A) Shift to the right by 4 units, reflect around the x-axis, then shift up by units; (B) Shift to the left by 4 units, reflect around the y-axis, then shift up by units; (C) Shift to the left by 4 units, reflect around the x-axis, then shift down by units; (D) Shift to the left by 4 units, reflect around the x-axis, then shift up by units; (E) none of them 19. Find the domain of f g if f(x) = x x+ and g(x) = 1 + x x 8

102 10 SOLUTIONS: 1. When x = 0, y = 6 = (0, ) When y = 0, 0 = x+6 x 0 = x + 6 x = 6 or ( 6,0). Since (1,18) is a solution of the function 18 = a(1 + )(1 ) 18 = a a = 6. (A) Function (B) Not function 4. C 5. f( ) = ( ) 5( ) + 7 = f(x) = (x)(x) (x) + 4 = 1x 4x Since < 1, f(x) = 7 x f( ) = 7 ( ) = f(x ) = (x )(x ) (x ) + 5 = (x x x + 4) x = x 11x (A) f() g() = 1 8 = 7 (B)f(x)g(x) = (5 x)(x + ) = 6x + 11x E x 0 15 x 5 x (, 5] 1. Denominator 0 x x 8 0 (x 4)(x + ) 0 (, ) (,4) (4, ) 1. x + 0 and x + 0 (, ) 14. D, E 15. f( x) = ( x) 4 + ( x) = x 4 + x = f(x) 16. y = (x + ) y = x D 19. ( f x ) (x) = x g x 1+ x 8 x, x 0, and x 8 x 8 x 0, x 8 0, and 1 + Then, the domain of f is all real numbers except,4,8 g x x x, x 0, and x 8 0 x, 8, 4

103 10 MORE PRACTICE PROBLEMS FOR EXAM 1. Identify the x-intercept and y-intercept of the graph of each function. (A) f(x) = x+6 (B) y = x 7x 4x x. Find the domain of the following functions (A) f(x) = x x + 4 (C) f(x) = x x 4. (B) f(x) = 8 x. (D) f(x) = x x 1 (E) f(x) = 5x 4 (F) f(x) = x x +x 6.. Decide the symmetry (Draw graph and mention about symmetry) (A) y = x + 5 (B) y = x x +1 (C) y = x + (D) y = x (A) x int: 6 ; y int: (B) x int: 1, 4,0; y int: 0 (A) (, ) (B) (, 4] (C) (, 4) (4, ) (D) [0,1) (1, ) (E) [ 4 5, ) (F) (, ) (,) (, ) (A) y axis (B) Origin (C) None (D) x axis 4. Decide whether each is a function or not. (A) x y = 4 (B) y = x x+ (C) y = x (D) y = x + (A) Not a function (B) Function (C) Function (D) Not a function (E) Not a function (F) Function (E) y = x + (F) y = x 5. Determine whether the function is even/odd/neither. (Draw graphs) (A) f(x) = x + 4 (B) f(x) = x 9x (C) f(x) = x 6 + 5x + 1 (D) f(x) = x x Find the following functions and state the domain of each using f(x) = x + 1 and g(x) = x : (A) (f + g)(x) (B) (f g)(4) (C) (fg)(x) (D) ( f g ) () (A) Neither (B) Odd (C) Even (D) Odd (A) x + x 1 (B) (C) 6x 4x + x (D) Let f(x) = x x + 5. Find the following. (A) f( ) (B) f(x) (C) f(x ) (D) f(x) (A) (B) 18x 9x + 5 (C) x 11x + 19 (D) x + x 5 x 8. Evaluate f () if f(x), x < 1 = { x + 1, x Evaluate f( ) if f(x) = { x + 1, x < 1, 1 x 1. x, x > If the graph of the function y = a(x + 5)(x ) passes through the point (1,4), then find the constant a a = 4

104 The graph of y = x is shifted right by units, reflected across the x-axis, and shifted up 5 units. Write the resulting equation. 1. The graph of y = x is shifted left by 5 units, reflected across the y-axis, and shifted down 4 units. Write the resulting equation. 1. The graph of y = x is shifted left by units, reflected across the x-axis, and shifted up 6 units. Write the resulting equation. 14. Describe the transformations that have been applied to obtain the function from the given base function : y = 4 x + ; y = x y = (x ) + 5 y = ( x + 5) 4 y = x Shift right units, vertically stretch by a factor of 4, and then shift up units 15. For the graph to the right, (A) Is it a function? (B) If so, find the domain and range. (C) f( 4) =? (D) Find the intercepts. (E) Find the local maximum (F) Find the decreasing interval(s) (A) Function (B) D: [ 6,5] R: [ 4,4] (C) f( 4) = (D) x int:,,5 y int: (E) 4 at x = (F) (,4) 16. Show that f(x) = x 4 5x + 7 is an even function algebraically f( x) 17. Which of the following statement is true? = ( x) 4 5( x) + 7 = x 4 5x + 7 = f(x) (E) (A) Its domain is [, 1] (B) Its range is [ 4,4] (C) It is an odd function. (D) It has a local maximum at a point (4,0) (E) It has a local minimum at a point (, ). 18. Find f(4) if f ( x x 1 ) = 1 4 x x + 5 f(4) =

105 105 UNIT. LINES AND QUADRATIC FUNCTIONS LECTURE -1 LINES Video 1) Find the slope and y-intercept of a given equation ) Find the equation of a line given two points ) Find vertical and horizontal lines LINE: Let (x 1, y 1 ) and (x, y ) be two points on a line. Slope of the line y Q(x, y ) m = y y 1 x x 1 = f(x ) f(x 1 ) x x 1 y 1 P(x 1, y ) x 1 Run = x x 1 Slope = m = y y 1 x x 1 x Rise = y y 1 Line equation: When slope is defined, ax + cy = d (general form) y = m(x x 1 ) + y 1 (point-slope form) y = mx + b (point-y intercept form) When slope is undefined, x = x 1 (Vertical line) Vertical line: x = a The other lines: y = mx + b where m is slope and b is its y intercept. y y y y x a x b y = mx + b x b y = b x b y = mx + b Slope is undefined m > 0: it is increasing m = 0: it is horizontal m < 0: it is decreasing

106 106 EXAMPLE 1 YOUR TURN 1 Find the slope and y-intercept of a line x y = 5 Find the slope and y-intercept of a line (A) 4x + y = 9 First, solve for y x y = 5 y = 5 x y = 5 + x y = x 5 Then, (B) 5x y = 7 The slope (the coefficient of x) is and y-intercept is 5. Since its slope is positive, the graph is increasing. EXAMPLE YOUR TURN Find the slope and y-intercept of a line y = Find the slope and y-intercept of a line y = 0 Since y = 0 x +, the slope is 0 and y-intercept is EXAMPLE YOUR TURN Find the slope and y-intercept of a line x = 0 Find the slope and y-intercept of a line x + 5 = 0 It does not have y; it is a vertical line which contains (,0) the slope is undefined and there is no y-intercept

107 107 EXAMPLE 4 YOUR TURN 4 Find the equation of a line that its slope is and it Find the equation of a line that its slope is 5 and it contains a point (,1). contains a point (,). And draw its graph. We use that y = m(x x 1 ) + y 1 m =, x 1 = y 1 = 1 y = (x ( )) + 1 = (x + ) + 1 = x = x + 7 Therefore, it is y = x + 7 EXAMPLE 5 YOUR TURN 5 Find the equation of a line that it contains a point (1, 4) Find the equation of a line that it contains a point (, ) and it is horizontal. and it is horizontal and draw its graph. Since it is horizontal, its slope is zero: m = 0 y = 0 (x 1) 4 y = 4 EXAMPLE 6 YOUR TURN 6 Find the equation of a line that it contains a point (1, 4) Find the equation of a line that its x intercept is and it is and it is vertical. vertical and draw its graph. Since it is vertical, its formula is x = x 1 x = 1

108 108 EXAMPLE 7 YOUR TURN 7 Find the equation of a line that contains points (, ) and (, ) Find the equation of a line that contains points ( 1, 6) and (, 4) 1) Find the slope Let (x 1, y 1 ) = (, ) and (x, y ) = (, ). m = y y 1 ( ) () = x x 1 () ( ) = 5 4 ) Find the equation of the line y = 5 (x ( )) + 4 = 5 (x + ) + 4 = 5 10 x = 5 4 x + 1 Therefore, y = 5 4 x + 1 EXAMPLE 8 YOUR TURN 8 Find the equation of a line that contains points (, 4) and (, 1) Find the equation of a line that contains points ( 1, ) and (4, 5) 1) Find the slope Let (x 1, y 1 ) = (, 4) and (x, y ) = (, 1). m = y y 1 ( 1) ( 4) = x x 1 () ( ) = 5 ) Find the equation of the line y = (x ( )) 4 5 = (x + ) 4 5 = 5 x = 14 x 5 5 Therefore, y = 14 x 5 5

109 109 EXAMPLE 9 YOUR TURN 9 Find the equation of a line that contains points Find the equation of a line that contains points (5, 4) and (5, 1) (,) and (,5) Since two points have same x value, it is a vertical line; x = 5 EXAMPLE 10 YOUR TURN 10 Find the equation of a line that contains points Find the equation of a line that contains points (, 4) and (, 4) ( 1, 5) and ( 4,5) Since two points have same y value, it is a horizontal line; y = 4 EXAMPLE 11 YOUR TURN 11 Find the equation of a line that contains a point (1, 4) Find the equation of a line that contains a point (, 1) and its x intercept is. and its y intercept is. It contains (1, 4) and (,0) 1) Find the slope Let (x 1, y 1 ) = (1, 4) and (x, y ) = (,0). m = y y 1 (0) ( 4) = x x 1 () (1) = ) Find the equation of the line y = (x 1) 4 = x 4 = x 6 Therefore, y = x 6

110 110 EXAMPLE 1 YOUR TURN 1 A truck rental company rents a moving truck for one week A truck rental company rents a moving truck for one day by charging $9 plus 5 cents per mile. by charging $150 plus 6 cents per mile. (A) Let x be the number of miles driven. Express the cost (A) Let x be the number of miles driven. Express the of renting a moving truck as a function of x. cost of renting a moving truck as a function of x. (B) What is the cost of renting the truck for one week if (B) What is the cost of renting the truck for one day if the truck is driven 110 miles? the truck is driven 0 miles? (C) How many miles can you travel in a week for $500? (C) How many miles can you travel in a day for $450? Let y be the cost and x be driven miles: (5 cents (= $0.05) per miles) x = driven miles y = cost 0 $9 1 $9 + $0.05 = 9.05 $9 + $ 0.05 = 9.10 This is a linear relation. The slope is m = y = 0.05(x 0) + 9 y = 0.05x + 9 = 0.05 When it drove 110 miles: x = 110 y = 0.05(110) + 9 = $ 4.5 When the cost is $500; y = = 0.05x = 0.05x x = 471 = 940 miles 0.05

111 111 EXAMPLE 1 YOUR TURN 1 Tom purchases a copy machine for $875. After 5 years, the machine must be replaced. Write a linear equation giving the value V of the equipment for 5 years it will be in use. A BAC purchased exercise equipment worth $1,000 for the new campus fitness center. The equipment has a useful life of 8 years. The salvage value at the end of 8 years is $,000. Write a linear equation that describes the book value of the equipment each year. Let x be the number of year. Number of year Value of equipment 0 $875 5 $0 Slope = = 175 and V-intercept is 5 V = 175x + 5 EXAMPLE 14 YOUR TURN 14 Suppose that the quantity supplied S and quantity demanded D of T-shirts are given by S(x) = x D(x) = x, where x is the price of a T-shirt. (A) Find the equilibrium price for a T-shirts. (B) What is the equilibrium quantity? The demand and supply equations for a new type of personal digital assistant are S(x) = x D(x) = x, where x is the number of units. Find the equilibrium number of digit assistant for this market. (A) Equilibrium price means x values such that supply is equal to demand; x = x x = = $16 (B) We want to find the number of supply (or demand) at the equilibrium price $16. S(16) = (16) = 600

112 11 EXAMPLE 15 EXAMPLE 16 Find the value of a in the figure if the area of the shaded Find the quadrant on which a line y = ax + b does not lie region is 1. if a line y = bx a is in the figure. y = ax + 4 y-axis y-axis x-axis y = bx a x-axis

113 11 PRACTICE PROBLEMS 1. Find the slope and its y-intercept (A) x y = 5 (B) 4y 5x = (C) 5x + 6y = 1 (D) 4x = y + 4. Find the equation of straight line (linear function) such that satisfies the following (A) slope 4 and contains a point ( 1, ) (B) slope and contains a point (,1) (C) Horizontal; contains a point (,6) (D) Vertical; contains a point (, 4) (E) x-intercept 5; contains a point ( 1,) (F) contains points (, 1),(6, 4) (A) m = b = 5 (B) m = 5, b = 1 4 (C) m = 5, b = 6 (D) m = 4, b = 4 (A) y = 4x + 7 (B) y = x 5 (C) y = 6 (D) x = (E) y = 1 x + 5 (F) y = x (G) contains points (4, ),(, 6). (I) contains points (, 1) and (,4) (K) x-intercept and y-intercept 4 (H) contains points (, 1),( 1, 6). (J) has x-intercept 4 and y-intercept 6 (L) y-intercept ; contains a point ( 1,) (G) y = x + (H) y = 5x 11 (I) x = (J) y = x 6 (K) y = x + 4 (L) y =. Find the constant a if a line x + ay = contains a point (1,) a = 1 4. Find the positive number a if the area of a right triangle bounded by ax + y = 6, x-axis, and y-axis is Find the constant k if three points A(1, k), B(k, k ) and C( k + 1, k + 4) are on the graph of a line L. 6. Find a point A(a, b) if A is on the graph of (m + )x + (m + 5)y + m + = 0 for any real numbers m. 7. Find the equation of all points which have equal distance from two points P(,) and Q(6,1) 8. The price of a new phone is $165. Its value decreases by $ 6.75 each year. Find the phone value after 17 years. a = 8 k = (1, 1) y = x 6 $ Tom and his sister, Mary, realizes that their combined ages equal their father s age in 016. If Tom is twice Mary s age and Sam is 69-year-old, what are Tom and Mary s ages in 016? Tom: 46-year-old Mary: -year-old

114 Tom purchases a copy machine for $875. After 5 years, the machine will have to be replaced. Write a linear equation giving the value V of the equipment for 5 years it will be in use. 11. The cost of a school banquet is $95 plus $15 for each person attending. Write an equation that gives total cost as a function of the number of people attending. What is the cost for 77 people? 1. In 1980 the average price of a home in Brainerd County was $97,000. By 1986 the average price of a home was $109,000. Write a linear model for the price of a home, P, in Brainerd County as a function of the year, t. Let t = 0 correspond to A car rental agency charges a weekly rate of $50 for a car and an additional charge of 15 cents for each mile driven. How many miles can you travel in a week for $610? V = 175x y = 15x + 95, $150 P = 000t miles 14. If a large factory sells its new gadgets for $5 each, it can sell 1050 per month, and if it sells the same gadgets for $9, it will sell 900 per month. Assuming the relationship between price and sales is linear, predict the monthly sales of gadgets to the nearest whole number if the price is $ An office has two envelope stuffing machines. Machine A can stuff a batch of envelopes in 5 hours, while Machine B can stuff a batch of envelopes in hours. How long would it take the two machines working together to stuff a batch of envelopes? $ hr 16. Mary starts walking along a path at 4 mi/h. Two hours after Ben leaves, his sister Amanda begins jogging along the same path at 8 mi/h. How long will it be before Amanda catches up to Ben? hours 17. A town s population has been growing linearly. In 004, the population was 6,00. By 009, the population had grown to 8,100. Assume this trend continues. (A) Predict the population in 01. (B) Identify the year in which the population will reach 15,000. (A) 960 (B) 08

115 115 LECTURE - RELATION OF TWO LINES Video 1) Parallel and perpendicular lines ) Linear application SPECIAL RELATION OF TWO LINES: y = m 1 x + b 1 x = c 1 x = c Parallel : Same slopes parallel : m 1 = m y = m x + b parallel : undefined slopes m 1 = m if m 1 and m are defined x = c 1,x = c if m 1 and m are undefined x = c Perpendicular y = m 1 x + b 1 y = m x + b perpendicular : undefined slope and zero slope perpendicular : m 1 = 1 90 m y = b m 1 = 1 m if m 1 0 and m 0 one is vertical and the other is horizontal EXAMPLE 1 YOUR TURN 1 Which of the following line is parallel to x + y = 5? Which of the following line is parallel to x y =? (A) 4y = x 8 (A) 4y = x 8 (B) y = x + 5 (B) y = x + 5 (C) y = x + (C) y = x + (D) x y = 5 (D) y = 1 x The line x + y = 5 (y = x + 5) has slope. Since the relation is parallel, parallel lines have same slope. (A) 4y = x 8 y = 1 x (B) y = x + 5 (C) y = x + (it is parallel to x + y = 5) (D) x y = 5 y = 1 x 5

116 116 EXAMPLE YOUR TURN Find the equation of a line that is parallel to x y = 7 Find the equation of a line that is parallel to x 4y = 4 and contain ( 1, ). and contains a point (, ). Step 1: Solve for y x y = 7 y = x + 7 y = x + 7 Step : Our line is parallel (same slope) to y = x 7 The slope is and it contains ( 1,) y = (x ( 1)) + = 11 x + EXAMPLE YOUR TURN Find the equation of a line that is parallel to x = 5 and Find the equation of a line that is parallel to y = 5 and contains a point (, 7). contains a point ( 4, ). Since it is parallel to a vertical line x = 5, our line is also a vertical line (undefined slope) and contains (,7). So, our line is x =.

117 117 EXAMPLE 4 YOUR TURN 4 Which of the following line is parallel to x + y = 5? Which of the following line is parallel to x y =? (A) 4y = x 8 (A) 4y = x 8 (B) y = x + 5 (B) y = x + 5 (C) y = x + (C) y = x + (D) x y = 5 (D) x y = 5 The line x + y = 5 (y = x + 5) has slope. Since the relation is parallel, the perpendicular line to the given line has + 1. (A) 4y = x 8 y = 1 x (B) y = x + 5 (C) y = x + (D) x y = 5 y = 1 x 5 to x + y = 5) (it is perpendicular EXAMPLE 5 YOUR TURN 5 Find the equation of a line that is perpendicular to Find the equation of a line that is perpendicular to a line x y = 7, contain ( 1, ). 5x y = 4 and contains a point (5, 1). Step 1: x y = 7 y = 7 x y = 7 x y = x 7 The slope of the given line is. Step : Our line is perpendicular to y = x 7 The slope is and it contains ( 1,) y = (x ( 1)) + = x + Then, y = x +

118 118 EXAMPLE 6 YOUR TURN 6 Find the equation of a line that is perpendicular to Find the equation of a line that is perpendicular to a line x + 4y = 5 and contains a point ( 1, ). a line x + y = 7 and contains a point (, ). Step 1: x + 4y = 5 4y = 5 x y = x y = 4 x The slope of the given line is 4. Step : Our line is perpendicular to y = 4 x The slope is + 4 and it contains ( 1,) y = 4 (x ( 1)) + = 4 10 x + Then, y = 4 10 x + EXAMPLE 7 YOUR TURN 7 Find the equation of a line that is perpendicular to a Find the equation of a line that is perpendicular to a line line y =, contain a point (5, 9). x = 1 and contains a point (1, ). Step 1: Since y = is a horizontal line. The line is vertical which is of the form x = x 1 Step : Find the value of x 1 x = 5 The linear equation is x = 5

119 119 DEFINITION: 1) Let y = f(x) be a function on the interval [a, b]. The average rate of change of f from a to b is f(b) f(a) b a (it means that the slope of a line which passes through (a, f(a)) and (b, f(b)) is the average rate of change of f in the interval [a, b]) ) If f(x) = mx + b is a linear function, the average rate of change of f on any interval is constant m which is given by the slope of f. EXAMPLE 8 YOUR TURN 8 Find the average rate of change of f(x) = x x + 1 in the given interval. Find the average rate of change of f(x) = x + 1 in the given interval. (A) When the interval is [ 1,], (A) When the interval is [ 1,], The average rate of change of f in [ 1,] is f() f( 1) ( 1) = (6) = 1 (B) When the interval is [0,] (B) When the interval is [0,], The average rate of change of f in [0,] is f() f(0) (0) 10 (1) = =

120 10 EXAMPLE 9 YOUR TURN 9 Determine whether the following function is linear or not. Determine whether the following function is linear or not. x y x y If the average rates of change in any interval are same, we can say that the function is linear. x y Δy average rate of change Δx (1) (4) ( 1) ( ) = ( ) (1) (0) (1) = ( 5) ( ) (1) (0) ( 8) ( 5) () (1) = = In this case, the average rates of change in any interval are. It is a linear function.

121 11 PRACTICE PROBLEMS 1. Let A(,) be a point. Find the line which contains A and (A) is parallel to y-axis (B) is perpendicular to y-axis (A) y = (B) x =. Find the equation of straight line (linear function) such that is parallel to a line (A) y = x + and contains (0,5) (B) x 8y = ; contains (1, ) (C) x + y = 1; contains (1, 1) (D) x y = 5; contains (, ) (A) y = x + 5 (B) y = 8 x (C) y = x + 1 (D) y = 1 x + (E) y = 5 x 1 (F) y = 7 (E) x + 5y = 4 and contains ( 5,). (F) y = and contains ( 1,7). Find the equation of straight line (linear function) such that is perpendicular to a line (A) y = x + and contains (1, ). (B) x y = 1 and contains (1, ). (C) x + y = ; contains (, 1) (D) x y = 5; contains (, 1) (E) x + 4y = 5 and contains (, 1). (F) x = and contains (, 4). (A) y = 1 x + 5 (B) y = x + 5 (C) y = 1 x (D) y = x + 4 (E) y = 4 x 5 (F) y = 4 4. Find the constant a if two lines y = a x + 1 and y = x + a are parallel to each other a = 1 5. Find the constant a if two lines y = ax + a and y = (a )x + are perpendicular to each other a = 1 6. Find the perpendicular bisector of a segment PQ where two points P(0,4) and Q(,1) y = x Find the distance of two intersection points of y = x + and (x + ) + (y ) = Find all k if x + y + k = 0 is a tangent line to (x 1) + (y + ) = 10 k = 18, 9. Find a constant k if a line y = kx + 6 is tangent to a circle x + y = 9. k = ± 10. Find the radius r if a circle x + y = r is tangent to a line x + y 5 = 0. r = Find the equation of a line such that it is tangent to a circle x + y = 9 and it is parallel to a line y = x + y = x + 10 y = x Find the tangent line to a circle x + y = 5 such that it is perpendicular to x + y + 1 = 0 y = x ± 5 1. Find the tangent lines to a circle (x + 1) + (y 4) = 9 such that its slope is y = x + 6 ± 5

122 1 LECTURE - SYSTEM OF LIEAR EQUATIONS Video 1) Elimination Method ) System application (1) ) System application () MEANING OF SOLVING THE SYSTEM OF EQUATION: x 4y = 5 Solve { means Find the intersection point of the two linear lines x 4y = 5, x + y = 7 x + y = 7 There are three types of systems of linear equations in two variables, and three types of solutions. An independent system has exactly one solution pair (x, y). The point where the two lines intersect is the only solution. An inconsistent system has no solution. Notice that the two lines are parallel and will never intersect. A dependent system has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations. Consistent system Inconsistent system Independent Dependent parallel lines one intersection same line No solution One solution (one intersection) Infinite many solutions (same line) EXAMPLE 1 YOUR TURN 1 x + y = 7 x y = 7 Consider {. Determine whether (, ) is Consider {. Determine whether (,4) is a x y = 0 x + y = 0 a solution of the system: solution of the system Substituting (, ) in x + y = 7 gives ( ) + ( ) = 7 7 = 7 : True Substituting (, ) in x y = 0 gives ( ) ( ) = 0 0 = 0 : True Therefore, it is a solution.

123 1 EXAMPLE x y = 1 Solve the system of linear equations { x 5y = EXAMPLE Step 1: find LCM of the x coefficients (or the y coefficients): 6 = LCM(,) Step : Make the x coefficients 6, 6 by multipling some constants. (x y = 1)() (x 5y = )( ) Step : Add two equations; 6x 9y = + 6x + 10y = 6 y = 6x 9y = 6x + 10y = 6 Step 4: Find the value of x by using one of equation and y =. Substituting y = in x y = 1 gives x () = 1 x = 4 Step 5: Check your answer (4) () = 1 1 = 1 (True) (4) 5() = = (True) Therefore, the solution is {(4,)} 5x y = 17 Solve the system of linear equations { x + 5y = 4 (A) Step 1: find LCM of the y coefficients (or the x coefficients): 5 = LCM(1,5) Step : Make the x coefficients 10, 10 by multiplying some constants. (5x y = 17)(5) x + 5y = 4 Step : Add two equations; + 5x 5y = 85 x + 5y = 4 7x = 81 5x 5y = 85 x + 5y = 4 x = Step 4: find the value of y by using one of equation and x = Substituting x = in 5x y = 17 gives 5() y = 17 y = Step 5: Check your answer 5() ( ) = = 17 (True) () + 5( ) = 4 4 = 4 (True) Therefore, the solution is {(, )} 6 4 x y = 1 x 5y = (4,) x + 5y = 4 (, ) 5x + y = 9

124 14 YOUR TURN Solve the system of linear equations x + y = 4 x y = 5 YOUR TURN Solve the system of linear equations 4x y = 11 x + y = 5 YOUR TURN Solve the system of linear equations x y = 1 x 5y = YOUR TURN Solve the system of linear equations x y = 1 y = 5x 17

125 15 EXAMPLE 4 YOUR TURN 4 Solve the system of linear equations x + y = 6x + y = x y = 8 Solve the system of linear equations x y = 8 Step 1: find LCM of the x coefficients 6 = LCM(,6) Step : Make the x coefficients 6, 6 y multiplying some constants. (x + y = )( ) 6x + y = Step : Add two equations; 6x y = 6 + 6x + y = 0 = 9 6x y = 6 6x + y = It is a False statement. It means that two lines are parallel. Therefore, it has no solutions. EXAMPLE 5 YOUR TURN 5 Solve the system of linear equations x + y = 6x + y = 6 Solve the system of linear equations 10x 4y = 5x + y = 1 Step 1: find LCM of the x coefficients 6 = LCM(,6) Step : Make the x coefficients 6, 6 by multiplying some constants. (x + y = )( ) 6x + y = 6 Step : Add two equations; 6x y = 6 + 6x + y = 6 0 = 0 6x y = 6 6x + y = 6 It is a True statement. It means that two lines are same. Therefore, it has infinite solutions. The solution set is {(x, y): x + y = }

126 16 EXAMPLE 6 YOUR TURN 6 A movie theater sells tickets for $8 each, with senior for $6. One evening, the theater took in $580 in revenue. Suppose that 55 tickets were sold that evening. Find the number of discount tickets sold on that day. A movie theater sells tickets for $9.00 each and seniors for $6.50. One evening the theater sold 600 tickets and took in $ in revenue. How many of seniors tickets were sold? Step 1: Define variables Let x = the number of sold discount tickets (each $6) Let y = the number of sold regular tickets (each $8). Step : Use the information to make a table: The number of tickets Revenue $6 ticket case x 6x $8 ticket case y 8y Total 55 $580 Step : Set up the system of linear equations ticket number: x + y = 55 Revenue 6x + 8y = 580 Step 4: Solve the system (x + y = 55)( 8) 6x + 8y = 580 8x 8y = 400 6x + 8y = x 8y = 400 6x + 8y = 580 x = 60 x = 10 Step 5: Find the answer: We want to find the value of x Therefore, the number of discount tickets sold is 10.

127 17 EXAMPLE 7 YOUR TURN 7 A total of $4,000 is invested in two bank accounts that pay 4 % and 6 % interest per year. If the total interest received for one year was $1,170, how much was invested at 6 %? A bank loaned out $50,000. Part of the money earned 10% per year, and the rest of it earned 6% per year. If the total interest received for one year was $,500, how much was loaned at 10%? Step 1: Define variables Let x = the amount of money invested at 6 % account Let y = the amount of money invested at 4 % account Step : Use the information to make a table ( 5 percent of A is ( 5 ) A = 0.05A) 100 Amount of money Interest 6% account x 0.06x 4% account y 0.04y Total 4, Step : Set up the system of linear equations Total amount money: x + y = 4000 Interest 0.06x y = 1170 Step 4: Solve the system (x + y = 4000)( 0.04) 0.06x y = x 0.04y = x y = x 0.04y = x y = x = 10 x = 10,500 Step 5: Find the answer: We want to find the value of x Therefore, the amount of money invested at 6% account is $10,500

128 18 YOUR TURN Tift Church held its annual spaghetti supper and fed a total of 51 people. They charged $5.5 for adults and $ for children (1 and under). If they took in a total of $,141, find how many adults attended the supper. YOUR TURN A bank loaned out $70,000. Part of the money earned 10% per year, and the rest of it earned 5% per year. If the total interest received for one year was $5,65, how much was loaned at 10%?

129 19 EXAMPLE 8 YOUR TURN 8 An airplane can fly 000 miles in 5 hours into the headwind. With this same wind, the airplane can fly the same distance in 4 hours. Find the average wind speed and the average speed of the airplane. With a tail wind, a small Piper aircraft can fly 600 miles in hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the average wind speed and the average airspeed of the Piper. Step 1: Define variables Let x = the average speed of the airplane Let y = the average wind speed. Step : Use the information to make a table Into the head wind Other direction Actual speed Distance(miles) Time in air (hours) x y x + y Step : Set up the system of linear equations We know a relation between distance, speed and time: speed = Distance Time Speed with head wind x y = Speed with tail wind x + y = Step 4: Solve the system x y = 400 x + y = 500 x = 900 x = 450, y = 500 x = 50 Step 5: Find the answer: The average speed of the airplane is 450 mi/hr. and The average wind speed is 50 mi/hr.

130 10 EXAMPLE 9 EXAMPLE 10 A merchant wishes to mix candy worth $5 per pound with Thirty liters of a 40% acid solution is obtained by mixing candy worth $ per pound to get 60 pounds of a mixture a 5% solution with a 50% solution. How much of each that can be sold for $ per pound. How many pounds of solution is required to obtain the specified concentration $5 type of candy should be used? of the final mixture?

131 11 EXAMPLE 11 YOUR TURN 11 Solve the system of linear equations { x 1 y = x + y = 7 1 Solve the system of linear equations { x 1 y = 5 x y = 1 1 By (the first equation)(), { x y = 6 x + y = 7 By adding two equations, 1 6 x = 1 x = 1 ( 6 1 ) = 6 By using the second equation and x = 6 (6) + y = 7 y = 7 4 = The solution set is {(6,)} EXAMPLE 1 YOUR TURN 1 Solve the system of linear equations { x x 1 y+ = 1 y+ = Solve the system of linear equations { 8 6 x x+ y 5 = y 5 = 1 Let A = 1 x 1 and B = 1 y+ By (second equation)(), By adding two equations, 6A + 10B = 1 { 4A 5B = 6A + 10B = 1 { 8A 10B = 6 14A = 7 A = 7 14 = 1 By using the first equation and A = 1 6 ( 1 ) + 10B = 1 10B = B = 1 5 Then x 1 = 1 = x = 1 A y + = 1 = 5 y = B The solution set is {( 1,)}

132 1 PRACTICE PROBLEMS 1. Solve the system of linear equations. (A) (C) (E) (G) (I) (K) x + y = 10 x y = 6 x y = 5 x + y = 9 5x 4y = 11 x + y = 11 x = y x + y = 1 5x y = 10 4x + y = 8 0.x + 0.4y = x + 0.5y = 0.1 (B) (D) (F) (H) (J) (L) x + y = 4 4x y = 10 x y = 10 4x + 7y = 5 y = x 1 5x y = 1 5x + 4y = 1 4y = x + 15 x + y = 1 x = 4 0.1x + 0.y = x 0.1y = 0.04 (A) (8,) (B) ( 1,) (C) (, ) (D) (8, 1) (E) (1,4) (F) (,) (G) (0.,0.1) (H) (1,4) (I) (,0) (J) (4,) (K) ( 1,1) (L) (4,) (M) (,1) (N) (,5) x y = (M) x y = 5 5. Solve the system of linear equations. (A) (C) x y = 8 x y = 8 x y = 4 6x + 9y = 1 (N) (B) (D) 1 x y = 7 1 x 1 4 y = x 4y = 5x + y = 1 4x 6y = 8 6x + 9y = 6 (A) (B) {(x, y): 10x 4y = } (C) {(x, y): x y = 4} (D). Solve the system of equations: (A) (C) x + 6 y = 1 4 x y = 8 6 x+ = y = x+ y 5 x y = 4 (E) x y = 0 (B) (D) (F) x + y = 4 x 5 y = = 10 x 1 y 8 9 = 5 x+1 y 6 5x y = x 5y = 5 (A) (, ) (B) ( 7, 11 5 ) (C) (, 1) (D) (,) (E) (, ) (F) ( 10, ) (G) (,) (H) (,1) (G) x+1 + y = 5 x y+1 = 19 (H) 4 x y = 1 x 9 y = 1

133 1 4. Solve the system of equations (A) x + y = 6 x + 5y = 100 (B) x + y = 18 x + y = (A) (5,1), ( 5,1), (5, 1), ( 5, 1) (B) (1, 4), (1,4), ( 1,4), ( 1, 4) 5. At a concert, 7 adult tickets and 4 child tickets cost $ whereas 5 adult tickets and child ticket cost $169. Find the price of an adult ticket. $ 6. A movie theater sells tickets for $9.00 each and seniors for $5.5. One evening the theater sold 600 tickets and took in $495.5 in revenue. How many of seniors ticket were sold? A merchant wish to mix candy worth $5 per pound with candy worth $ per pound to get 60 pounds of a mixture that can be sold for $ per pound. How many pounds of $5 type of candy should be used? 8. A bank loaned out $100,000. Part of the money earned 10% per year, and the rest of it earned 6% per year. If the total interest received for one year was $7,000, how much was loaned at 10%? 9. Mary wants to mix a drink containing 40% juice with a drink containing 10% juice to produce 10 gallons of a drink containing 5% juice. How much of each drink should she use? 10. How much of the 8% acid solution should be added to a 14% acid solution to get 00 g of a 10 % acid solution? 0 pounds $5,000 Each 5 gallons 00 g 11. The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number? 4 1. The larger of two numbers is 5 more than twice the smaller. If the smaller is subtracted from the larger, the result is 1. Find the numbers. 7 and An airplane flying with the wind can cover a certain distance in hours. The return trip against the wind takes.5 hours. How fast is the plane and what is the speed of the air, if the one-way distance is 600 miles? 14. A boat traveled 10 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Airplane: 70 mph Wind: 0 mph Boat: 1 mph Water: 9 mph

134 14 LECTURE -4 QUADRATIC FUNCTIONS DEFINITION: A quadratic function is a function of the form f(x) = ax + bx + c where a, b, c are real numbers and a 0 whose graph is called a parabola. f(x) = ax + bx + c : standard form = a(x h) + k vertex form where a 0 1) The vertex of y = f(x) is the maximum or minimum point (h, k) on the graph of y = f(x). (h, k) = ( b a, b 4ac ) 4a ) The axis of symmetry of y = f(x) is a line through a shape so that each side is a mirror image. The axis of symmetry of y = f(x) is the vertical line x = b a a > 0 open up a < 0 open down axis symmetry vertex y-intercept x -intercept x -intercept Graph y-intercept x -intercept x -intercept open upward parabola; a > 0 vertex open downward parabola; a < 0 axis symmetry Max/Min Minimum k at x = h Maximum k at x = h Domain (, ) (, ) Range [k, ) (, k] ) x-intercepts (= real zeros) Two x-intercepts one x-intercept No x-intercepts Graph (a > 0 case) x axis x axis x axis Discriminant b 4ac > 0, b 4ac = 0, b 4ac < 0, x-intercept(s) b ± b 4ac a b a No real zeros

135 15 Video 1) Find the vertex of a quadratic function ) Quadratic Application (1) ) Quadratic Application () EXAMPLE 1 YOUR TURN 1 Use your calculator to find the vertex and Use your calculator to find the vertex and maximum/minimum. maximum/minimum. (A) y = x 8x + 5 (A) y = x 6x + (B) y = 5x + 10x + (B) y = x 1x + 7 EXAMPLE YOUR TURN Find the vertex of y = x + x + 4 without your Find the vertex of y = x 4x + without your calculator. calculator. 1) Vertex x value: ) Vertex y value: h = b a = ( ) = 1 k = ( 1 ) + ( 1 ) + 4 = 1 Therefore, the vertex is ( 1, 1 ) The standard form of this is y = (x 1 ) + 1

136 16 EXAMPLE YOUR TURN Determine the equation of a quadratic function that Determine the equation of a quadratic function that Its vertex is (1, 5) and y-intercepts is Its vertex(, );contains a point (1, 6) Since vertex is (1, 5) and y = a(x h) + k, y = a(x 1) + ( 5) Since y-intercept is, it passes through a point (0, ), y = a(x 1) + ( 5) = a(0 1) + ( 5) = a + ( 5) = a Therefore, y = (x 1) + ( 5) = (x 1) 5 EXAMPLE 4 YOUR TURN 4 Find the equation of a parabola shown in the figure. Find the equation of a parabola shown in the figure (1, ) We know that its vertex is (1, ) and the y-intercept is 1. Since its vertex is (1, ) and y = a(x h) + k, y = a(x 1) + Since y-intercept is 1, it passes through a point (0,1), y = a(x 1) + 1 = a(0 1) + 1 = a + 1 = a Therefore, y = (x 1) +

137 17 EXAMPLE 5 YOUR TURN 5 Find the intercepts of y = x 6x + Find the intercepts of y = x 6x 4 y-intercept: when x = 0 y = (0) 6(0) + = y-intercept is (0,) x-intercepts: when y = 0 0 = x 6x + b 4ac = ( 6) 4()( ) = 60 0 So, it has x-intercepts. x = ( 6) ± 60 () = 6 ± 15 4 = ± 15 x-intercept is ( + 15, 0), ( + 15, 0) EXAMPLE 6 YOUR TURN 6 Find the intercepts of y = x + 10x 17 Find the intercepts of y = x x + 5 y-intercept: when x = 0 y = (0) + 10(0) 17 = 17 y-intercept is (0, 17) x-intercept: when y = 0 0 = x + 10x 17 b 4ac = (10) 4( )( 17) = 6 < 0 So, it has No x-intercepts.

138 18 EXAMPLE 7 YOUR TURN 7 Consider f(x) = x + 8x + 5 Consider f(x) = x 6x + (A) Find the vertex and vertex form. (A) Find the vertex and vertex form. (B) Find the axis of symmetry (B) Find the axis of symmetry (C) Find the maximum or minimum of f (C) Find the maximum or minimum of f (D) Find the domain of f (D) Find the domain of f (E) Find the range of f (E) Find the range of f (F) Determine where f is increasing/decreasing (F) Determine where f is increasing/decreasing (G) Find the x-intercepts if any (G) Find the x-intercepts if any 1) First, find the x-value of vertex: h = b a = 8 () = Find the y-value of vertex: k = f( ) = ( ) + 8( ) + 5 = Then, its vertex is (, ) ) Since the coefficient of x is a =, the vertex form is f(x) = (x + ) ) The axis of symmetry is x = 4) Since the coefficient of x is a = > 0, 4 axis of symmetry x = y = x + 8 x (, ) It has minimum at x =. 5) Domain is (, ) 6) Since it is open up, its range is [, ) 7) The function f is decreasing on the interval (, ) and increasing on the interval (, ) 8) Find only real solution of x 8x + 5 = 0. x = 8 ± 8 4()(5) () = ± 6

139 19 EXAMPLE 8 YOUR TURN 8 Consider f(x) = x + x + Consider f(x) = 5x + 10x (A) Find the vertex and vertex form. (A) Find the vertex and vertex form. (B) Find the axis of symmetry (B) Find the axis of symmetry (C) Find the maximum or minimum of f (C) Find the maximum or minimum of f (D) Find the domain of f (D) Find the domain of f (E) Find the range of f (E) Find the range of f (F) Determine where f is increasing/decreasing (F) Determine where f is increasing/decreasing (G) Find the x-intercepts if any (G) Find the x-intercepts if any 1) First, find the x-value of vertex: h = b a = ( 1) = 1 Find the y-valuve of vertex: k = f(1) = (1) + (1) + = 4 The vertex is (1,4) ) Since the coefficient of x is a = 1, the vertex form is f(x) = (x 1) + 4 ) The axis of symmetry is x = h x = 1 4) Since the coefficient of x is a = 1 < 0, 6 4 (1, 4) y = x + x + 5 Axis of symmetry x = 1 It has maximum 4 at x = 1. 5) Domain is (, ) 6) Range is (, 4] 7) The function f is increasing on the interval (, 1) and decreasing on the interval (1, ) 8) Find the real solution of (x x ) = 0 (x )(x + 1) = 0 x = 1,

140 140 EXAMPLE 9 YOUR TURN 9 Consider f(x) = x + 6x + Consider f(x) = 4x 4x + 1 (A) Find the vertex and vertex form. (A) Find the vertex and vertex form. (B) Find the axis of symmetry (B) Find the axis of symmetry (C) Find the maximum or minimum of f (C) Find the maximum or minimum of f (D) Find the domain of f (D) Find the domain of f (E) Find the range of f (E) Find the range of f (F) Determine where f is increasing/decreasing (F) Determine where f is increasing/decreasing (G) Find the x-intercepts if any (G) Find the x-intercepts if any 1) First, find the x-value of vertex: h = b a = 6 () = 1 Find the y-valuve of vertex: k = f( 1) = ( 1) + 6( 1) + = 0 The vertex is (h, k) = ( 1,0) ) Since the coefficient of x is a =, the vertex form is f(x) = (x + 1) ) The axis of symmetry is x = h x = 1 4) Since the coefficient of x is a = > 0, 6 axis of symmetry x = ( 1,0) y = x + 6 x + It has minimum 0 at x = 1 5) Domain is (, ) 6) Range is [0, ) 7) The function f is decreasing on the interval (, 0) and increasing on the interval (0, ) 8) The x-intercept is 1

141 141 EXAMPLE 10 YOUR TURN 10 The length of a rectangle is 6 cm greater than its width. The length of a rectangle is cm greater than its width. The area of the rectangle is 7 square centimeters. Find The area of the rectangle is 6 square centimeters. Find the length of the rectangle. the length of the rectangle. W L = W + 6 The area of the rectangle is 7 cm and the length is 6 cm greater than its width: LW = 7 and L = W + 6 (W + 6)W = 7 W + 6W = 7 W + 6W 7 = 0 (W + 9)(W ) = 0 Since W > 0, W = cm and L = W + = 9 cm Therefore, its length is 9 cm

142 14 EXAMPLE 11 YOUR TURN 11 A ball is thrown vertically upward. Its height h(t) in A ball is thrown vertically upward from a 18-meter meter above the ground after t seconds is given by building. Its height h(t) in meter above the ground after t h(t) = 80t t seconds is given by (A) When will the ball reach maximum height? h(t) = 16t + 11t + 18 (B) What is the maximum height reached by the ball? (A) When will the ball reach maximum height? (C) How long will the ball be in flight? (B) What is the maximum height reached by the ball? (D) When does the ball hit the ground again? (C) How long will the ball be in flight? (E) When will the ball reach 1500 meters above the (D) When does the ball hit the ground? ground? (E) When will the ball reach 88 meters above the ground? (A) To find t at which h becomes maximum, we should find the vertex t-value of the vertex: b a = 80 ( 1) = 40 The ball reaches the maximum height after 40 seconds. (B) When it reaches the maximum height after 40 seconds. h(40) = 80(40) (40) = 1600 meters (C) We must find the time when the ball hit the ground (height is zero) 80t t = 0 t(80 t) = 0 t = 0, 80 So, it is in flight for 80 seconds. (D) hit the ground means the height is zero in (B) So, it hits the ground after 80 seconds. (E) Since the height is 1500 meters, 1500 = 80t t t 80t = 0 (t 0)(t 50) = 0 So, it reaches 1500 meters above the ground after 0 seconds and 50 seconds

143 14 EXAMPLE 1 YOUR TURN 1 The profit from selling local ballet tickets depends on the An object is thrown vertically upward from 48-foot tree ticket price. Using past receipts, we find that the profit and its distance h(t) in feet above the ground after t can be modeled by the function seconds is given by the formula P(x) = 15x + 600x + 60 h(t) = 4t + 48 where x is the price of each ticket. We want to find (A) When does the object hit the ground? the ticket price that gives the maximum profit and find (B) When will the object reach 1 meters above the that maximum profit. ground? We want to find x that give the maximum profit (Vertex) x = b a = 600 ( 15) = 0 P(0) = 15(0) + 600(0) + 60 = 6060 Then, the maximum profit is $6,060 at the ticket price $0 EXAMPLE 1 YOUR TURN 1 Find the maximum vertical distance d between the Find the maximum vertical distance between the parabola parabola and the line for the shaded region. and the line for the shaded region. 6 4 y = x + 4 x y = x 5 4 y = x + 4 x + 4 y = x + Consider h(x) = Top bottom = ( x + 4x + ) (x ) = x + x + 5 d is the vertex y-value of h(x) Vertex x-value: b a = ( ) = 4 d = ( 4 ) + ( 4 ) + 5 = 49 8

144 144 EXAMPLE 14 YOUR TURN 14 Tom has 100 meters of fencing to enclose a rectangular A farmer has 1600 yards of fence to enclose a rectangular field. field. What are the dimensions of the rectangle that (A) What is the maximum area? encloses the most area? What is the largest area of this (B) Find the dimensions of the rectangle that rectangular field? maximize the enclosed area. Step 1: Decide what value is max/min: Let x be the length and w the width. Let A be the area of the rectangle. Since the perimeter is 100, w x + w = 100 w = 50 x A = xw = x(50 x) = x + 50x x Step : Find the vertex: To find x at which A has maximal value, we should find the vertex x-value of the vertex: Then the maximal area A is b a = 50 ( 1) = 5 A = 5(50 5) = 65 Step : Answer the question: The maximum area is 65 m and the dimension is 5 m 5 m

145 145 EXAMPLE 15 YOUR TURN 15 Tom wants to construct a rectangular parking lot on land Mary plans to make a rectangular garden next to her bordered on one side by a river. It has 10 ft. of fencing house. If she uses 00 meters of fencing material by using that is to be used to fence off the other three sides. What her house as one side and fencing for the other three sides, should be the dimension of the lot if the enclosed area is find the maximum area of her garden. to be maximum? What is the maximum area? Step 1: Decide what value is max/min: river W W L Let L be the length and W the width. L + W = 10 L = 10 W Let A be the area of the rectangle. A = LW = (10 W)W = W + 10W Step : Find the vertex: To find W at which A has maximal value, we should find the vertex W- value of vertex: b a = 10 ( ) = 0 A -value of vertex: (0) + 10(0) = 1,800 Step : Answer the question: W = 0, L = 10 (0) = 60 The dimension of lot is 0ft - 60ft - 0ft. The maximum area is 1,800 ft.

146 146 EXAMPLE 16 EXAMPLE 17 A rain gutter is to be made of iron sheets that are 18 cm Taylor and Miranda are performing on a magic wide by turning up in the figure. Find the depth if it has dimension-changing stage that is 0 yards long by 15 the maximum cross section area. yards wide. The length is decreasing linearly (with time) at a rate of yards per hour, and the width is increasing linearly (with time) at a rate of yards per hour. When will the stage have the maximum area, and x x x 18 cm when will the stage disappear (have an area of 0 square yards)?

147 147 PRACTICE PROBLEMS 1. (a) graph each quadratic function by determining whether the graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Find maximum/minimum (A) f(x) = x + (B) f(x) = x + x (C) f(x) = 4x + 8x 1 (D) f(x) = x + 8x + (E) f(x) = x 10x + (F) f(x) = x + 1x + 5 (G) f(x) = 1 x 4x + 1 (H) f(x) = x + x (I) f(x) = x 1x + 1 (J) f(x) = x + x 1 (A) (a)(0,); x = 0; (0,); (, 0), (, 0) (b)(, ); (, ] (c)max (B) (a)( 1, 4); x = 1; (0, ); (, 0), (1,0) (b)(, ); [ 4, ) (c)min 4 (C)(a)(1,); x = 1; (0, 1); ( ±, 0) (b)(, ); (, ] (c)max (D)(a)(, 5); x = ; (0,); ( 4± 10, 0) (b)(, ); [ 5, ) (c)min 5 (E)(a)(5, ); x = 5; (0,); (5 ±, 0) (b)(, ); [, ) (c)min (F)(a)(,); x = ; (0,5); ( 6± 46, 0) (b)(, ); (, ] (c)max (G)(a)(6,0); x = 6; (0,1); (6, 0) (b)(, ); [0, ) (c)min 0 (H)(a) ( 1, 1 6 ) ; x = 1/; (0,0); (0, 0), (, 0) (b)(, ); (, 1 6 ] (c)max 1 6 (I)(a)(,0); x = ; (0,1); (, 0) (b)(, ); [0, ) (c)min 0 (I)(a) ( 1, 0) ; x = 1/; (0, 1/); (1/, 0) (b)(, ); (, 0] (c)max 0. Find the maximum or minimum value of the quadratic function. (A) f(x) = (x ) + 8 (B) f(x) = (x + 4) 5 (C) f(x) = x + 10x 4 (D) f(x) = 4x 8x +. Find the equation of quadratic function for which: (A) Vertex is (, ); contains the point (1, 6) (B) Vertex is (4, ); contains the point (, 1) (C) Vertex is ( 1, ); contains the point (1, 14) (A) MAX 8 (B) MIN 5 (C) MAX 1 (D) MIN 1 (A) y = (x ) (B) y = 1 (x 4) (C) y = 4(x + 1) +

148 Find the equation of quadratic function such that (A) 5 (B) ( 1,) 4 (A) y = (x 1) (B) y = (x + 1) + (C) y = (x + 1)(x ) (D) y = (x + )(x ) (C) 4 (1, ) (D) (0,9) ( 1,0) (,0) 5 (0, 6) (,0) (,0) 5 5. Find the signs of a, b, and c if the graph of y = ax + bx + c is in the figure. (A) (B) (A) a > 0, b < 0, c < 0 (B) a < 0, b < 0, c > 0 6. Describe the transformations that have been applied to obtain the function y = x 10x + 6 from y = x x +. Shift right 4 units and shift down 1 unit. 7. Given the quadratic function f(x) = ax + bx + c in the figure. The area of the shaded triangle (whose edges are x-intercepts 0, 6 and the vertex of f) is 18. Find the constants a, b, and c. y = ax + bx + c vertex a =, b = 4, c = Find the equation of the quadratic function such that it passes through two points (,0) and (6,0) and it has maximum Find the maximum value of a if the graph of y = x + 6x + a does not be above the x-axis. 10. Find the constant a if the graphs of y = x x and y = a meets exactly at three points. a = 1 y = x + 4x + 1 7

149 The width of a rectangle is feet less than the length. If the area of the rectangle is 54 square feet, find the length of the rectangle. 9 ft. 1. Tom has 150 feet of fencing available to enclose a rectangular field. One side of the field lies along the highway, so only three sides require fencing. Find the largest area of the rectangular field. 1. Park has a 1000-foot roll of fencing and a large field. He wants to make four paddocks by splitting a rectangular enclosure in same size (as in the figure). What are the dimensions of the largest such enclosure? 81.5 square ft 5 pieces: 100 ft. and pieces: 50 ft. 14. A ball is propelled vertically upward with an initial velocity of 0 meters per second. The distance s (in meters) of the object from the ground after t seconds is s(t) = 5t + 0t. (A) When will the ball be 15 meters above the ground? (B) When will it strike the ground? (C) What is the maximum height? (A) 1, sec (B) 4 sec (C) 0ft 15. A person standing close to the edge on the top of a 51-foot building throws a ball vertically upward. The quadratic function s(t) = 16t + 64t + 51 models the ball s height above the ground, s(t), in feet, t seconds after it was thrown. (A) After how many seconds does the ball reach its maximum height? (B) What is the maximum height? (C) How many seconds does it take until the ball finally hits the ground? (A) (B) (C) sec 576 ft 8 sec 16. A football stadium holds 6,000 spectators. With a ticket price of $11, the average attendance has been 6,000. When the price dropped to $9, the average attendance rose to 0,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue? 17. A hockey team plays in an arena that has a seating capacity of 15,000 spectators. With the ticket price set at $14, average attendance at recent games has been A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by (A) Find a function that models the revenue in terms of ticket price. (B) What ticket price is so high that no one attends, and hence no revenue is generated? (C) Find the price that maximizes revenue from ticket sales. $1 (A) R = 500x 1000x (B) $.50 (C) $11.75

150 150 MATH 1111 COLLEGE ALGEBRA REVIEW FOR EXAM Work these specifically. Try them WITHOUT your notes or text. 1. Find the slope and y-intercepts of 5x y = 9. Find the equation of a line which contains two points (,) and (4, ). Find the equation of a line which is parallel to x + y = 8 and contains a point (,4) 4. Find the equation of a line which is perpendicular to x 4y = 7 and contains a point (, 1) 5. A car rental agency charges a weekly rate of $50 for a car and an additional charge of 5 cents per each mile driven. How many miles can you travel in a week for $610? 6. Solve the system of linear equations: 7. Solve the system of linear equations: 8. Solve the system of linear equations: 9. Solve the system of linear equations: x y = 10 4x + 7y = 5 15x 6y = 5x + y = 1 x + 11y = 4 x 5y = 9 5x + 6y = 0x 4y = A movie theater sells tickets for $9.00 each and seniors for $5.50. One evening the theater sold 600 tickets and took in $ in revenue. How many of seniors tickets were sold? 11. A bank loaned out $100,000. Part of the money earned 10% per year, and the rest of it earned 6% per year. If the total interest received for one year was $7,000, how much was loaned at 10%? 1. Find the vertex of the quadratic function: f(x) = x + 1x Decide whether f(x) = x 4x + 5 has maximum or minimum and find its max/min value.

151 Let f(x) = x 6x + 1. (A) Find the vertex of f. (B) Rewrite it in the standard form (vertex form) (C) Find the domain of f. (D) Find the range of f. (E) Find the axis of symmetry of f. (F) Find all x-intercepts of f if any. (G) Find the increasing interval of f. (H) Draw the graph of f(x). 15. Find the equation of a quadratic function such that its vertex is (4, ) and it contains a point (, 1). 16. A person standing close to the edge on the top of a 51-foot building throws a ball. The quadratic function h(t) = 16t + 51 models the ball s height above the ground, h(t), in feet, t seconds after it was thrown. How many seconds does it take until the ball finally hits the ground? 17. Tom has 10 feet of fencing available to enclose a rectangular field. One side of the field lies along the highway, so only three sides require fencing. Find the largest area 18. The length of a rectangle is 4 cm longer than its width. The area of the rectangle is 45 square centimeters. Find the length of the rectangle. 19. The height, in feet, from the ground of a ball thrown from an 800-foot building at t seconds after it is dropped is given by the formula h(t) = 16t + 80t At what time will the ball reach its maximum height? What is the maximum height?

152 15 SOLUTIONS 1. y = 5x + 9 y = 5 x + 9 Slope 5 ; y-int. Slope: ( ) 4 ( ) = 5 7 y = 5 (x ( )) + = 5 (x + ) y = 5 7 x 1 7. x + y = 8 y = x + 8 Since slope of our line is equal to slope of y = x + 8, our line is y = (x ) + 4. y = x x 4y = 7 y = 4 x 7 4 Since it is perpendicular to each other, slope of our line is 4 y = 4 (x ) + ( 1) (different sign and reciprocal) y = 4 x Mileage fee is $610 $50 = $60 Let x be the mileage and $0.05 per mile. 60 = 0.05x x = miles 6. 4x + 8y = 40 4x + 7y = 5 First equation: 15y = 15 y = 1 x ( 1) = 10 x + = 10 x = 8 The solution is {(8, 1)} 7. 15x 6y = 0 = 1: False statement 15x + 6y = 8. 6x + y = 8 6x 15y = 7 7y = 5 y = 5 no solutions. First equation:x + 11(5) = 4 x + 55 = 4 x = 51 = 17 The solution is {( 17,5)} 9. 0x + 4y = 1 0x 4y = 1 Infinite many solutions; {(x, y): 5x + 6y = } 0 = 0: True statement 10. x: number of $5.5 ticket, y: number of $9 ticket x + y = x + 9y = x 9y = x + 9y = x = x = x: amount in 10%, y: amount in 6% x + y = 100, x y = x 0.06y = x y = 7000 $5, x = (,7)

153 15 1. It has minimum at x = A: (, 7 ) B: y = (x ) 7 C: (, ) D: [ 7, ) E: the axis of symmetry is a line x =. F:x = ± 7 G: [, ) H: 5 4 ( ), y = a(x 4) 1 = a( 4) 1 = 4a = 4a a = 1 y = 1 (x 4) 16. hit the ground means that h = 0 0 = 16t t = 51 t = t = ± Since t represent time, t is positive. Therefore, it hit after = second. 17. fencing: 10 = L + W L = 10 W Area = LW = (10 W)W = 10W W Vertex W-value: b a = 10 ( ) = 0 Vertex Area: 10(0) (0) = 1800 The largest area is 1800 ft cm 19. Maximum time: b a = 80 ( 16) = 5 Maximum height h ( 5 ) = 16 (5 ) + 80 ( 5 ) = 900

154 154 MORE PRACTICE PROBLEMS FOR EXAM 1. Find the slope and its y-intercept of x y = 5. m = b = 5. Find the equation of straight line such that (A) contains the points (, ) and (6, 6). (B) has x-intercept and y-intercept 6. (C) is perpendicular to the line 4x + y = and containing the point (1, ). (D) is parallel to the line x 4y = 5 and containing the point (, 1). (A) y = 4 x (B) y = x + 6 (C) y = 1 11 x (D) y = 4 x 1. Find the linear equation which satisfies f( ) = 5 and f() =. f(x) = x A car rental agency charges a weekly rate of $150 for a car and an additional charge of 5 cents for each mile driven. How many miles can you travel in a week for $500? 7000 miles 5. Solve the system of linear equations. 4x y = 11 (A) { x + y = 5 x y = 5 (C) { x + y = 7 x + y = 1 (B) { x + y = 5 x y = 7 (D) { 9x y = 1 (A) (, 1) (B) (1, 7) (C) (D) {(x, y); x y = 7} 6. You inherit $10, 000 and will invest the money in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at 6% interest stock option if the total interest earned for the year is to be $900? 7. A movie theater sells tickets for $9.00 each, with seniors for $7.00. One evening the theater sold 600 tickets and took in $4760 in revenue. How many of seniors ticket were sold? 8. Grace Church held its annual spaghetti supper and fed a total of 15 people. They charged $5 for adults and $ for children (1 and under). If they took in a total of $590, find how many adults attended the supper. $6, Find the vertex form which is equivalent to f(x) = x + x. f(x) = (x + 1) 4 Problem 1~ 14: Consider the following function (A) Find the vertex and vertex form. (B) Find the axis of symmetry (C) Find the domain. (D) Find the range. (E) Find the maximum or minimum.

155 f(x) = x + 6x + 5 (A) (1,8) (B) x = 1 (C) (, ) (D) (, 8] (E) Max 8 at (1,8) 11. f(x) = x + x + (A) ( 1,1) (B) x = 1 (C) (, ) (D) [ 1, ) (E) Min 1 at ( 1,1) 1. f(x) = x + 5x + (A) ( 5, 7 4 ) (B) x = 5 (C) (, ) 1. Find the equation of a quadratic function such that its graph has the vertex at the point (4, ) and (, 1) is also a point on the graph of the function. (D) (, 7 4 ] (E) Max 7 4 at ( 5, 7 4 ) y = 1 (x 4) 14. An object is thrown vertically upward and its distance s(t) in feet above the ground after t seconds is given by the formula s(t) = 4t + 64t (A) Find its maximum distance above the ground. (B) When will it hit the ground again? 15. David has available 500 yards of fencing and wishes to enclose a rectangular area. Express the area A of the rectangle as a function of the width x of the rectangle. For what value of x is the area largest? 16. Tom has 160 feet of fencing available to enclose a rectangular field. One side of the field lies along the highway, so only three sides require fencing. Find the largest area of the rectangular field. 17. The length of a rectangle is cm longer than its width. The area of the rectangle is 6 cm square centimeters. Find the length of the rectangle. (A) 56 (B) 16 secs A = x(50 x) x = 15 yards 00 ft 9 cm 18. The observed bunny rabbit population on an island is given by the function p(t) = 0.4t + 10t where t is the time in months since they began observing the rabbits. (A) When is the maximum population attained? (B) what is the maximum population? (C) when does the bunny rabbit population disappear from the island? (A) After 16.5 months (B) (C) After 4 months

156 156 UNIT 4. SEVERAL FUNCTIONS LECTURE 4-1 POLYNOMIALS Video 1) Introduction to Polynomial functions ) Use Zeros to find a polynomial function POLYNOMIAL: Let f(x) = a n x n + + a 1 x + a 0 be a polynomial where a j s are constant and a n 0. (A) Degree: n (the number of x-intercepts can be at most n) (B) Leading term: a n x n (C) Leading coefficient: a n (D) f(x) is a smooth and continuous function (E) The real zeros of f(x) = the x- intercepts of f(x) = real roots of f(x) 5 5 y = (x + )(x ) EXAMPLE 1 YOUR TURN 1 Determine whether f(x) = 5 7x 5 + 8x is a polynomial. Determine whether f(x) = x 8 7x + 9 is a If it is, state the degree of the function. polynomial. If it is, state the degree of the function. Yes, it is a polynomial Leading term 7x 5 Leading coefficient 7 Degree 5

157 157 EXAMPLE YOUR TURN Determine whether f(x) = is a polynomial. If it is, x state the degree of the function. Determine whether f(x) = x is a polynomial. If it is, state the degree of the function. Since f(x) = x 1, it is not a polynomial. EXAMPLE YOUR TURN Find the degree and the leading coefficient of Find the degree and the leading coefficient of f(x) = (x )(x 1) (x + ) (A) f(x) = (x + )(x 1) (x 5) To find the leading term, we simplify f by using distributive law. But, it is too complicated, and we use the leading terms of each factors; f(x) = (x ) x (x 1) (x + ) (x) (x) = 54x 6 81x 5 90x x + 70x 44x 4 (B) f(x) = (5x )(x + 7) (x + 1) 4 The leading term of f is x x (x) = 54x 6 So, its degree is 6 and the leading coefficient is 54. EXAMPLE 4 YOUR TURN 4 Which of following is a factor of a polynomial f(x) = x 4 + 7x 6x 49x + 0? Which of following is a factor of a polynomial f(x) = 6x 5x + 19x + 0? (A) x 5 (B) x + 1 (A) x + 5 (B) x + (C) x + (D) x + (C) x + (D) x + 6 Since a linear factor is related with a x-intercept, f(5) = 160; f ( 1 ) = 47.5 f( ) = 0; f( ) = 84 Since x = is x-intercpet, (x + ) is a factor of f(x) So, the answer is (C).

158 158 DEGREE OF A POLYNOMIAL: maximum number of real zeros (x-intercepts) of a polynomial or number of all real/complex zeros (with multiplicity) of a polynomial MULTIPLICITY: If (x r) k is a factor of a polynomial function f and (x r) k+1 is not a factor of f, then r is called a zero of multiplicity k of f. (A) If k is even, the graph of f touches the x-axis at x = r (B) If k is odd, the graph of f crosses through the x-axis at x = r f( x) = 0.1 ( x + ) ( x ) cross 5 Touch x-axis EXAMPLE 5 YOUR TURN 5 Find all zeros and its multiplicity of the function and Find all zeros and its multiplicity of the function and decide whether it crosses or touches the x-axis at each decide whether it crosses or touches the x-axis at zeros. zero. (A) f(x) = x (x ) 8 (x + ) 5 f(x) = x (x + )(x x ) (x 4) We factor this completely; x (x + )((x )(x + 1)) ((x )(x + )) = x (x + )(x ) (x + 1) (x ) (x + ) = x (x + )(x ) 5 (x + 1) (x + ) Since f(x) = x (x + )(x ) 5 (x + 1) (x + ), it has five different linear factors. (B) f(x) = 5(x + x ) (x x 6) Factors x x + x x + 1 x + x-intercepts 0 1 Multiplicity 1 5 Cross/touch touch cross cross touch cross

159 159 EXAMPLE 6 YOUR TURN 6 Find a polynomial function f such that Find a polynomial function f such that Degree: Degree: 4 its zeros are,, and 5 and its zeros are 4, 1, 1 and and its leading coefficient is 5. its leading coefficient is. f(x) = 5 (x + )(x )(x 5) LC EXAMPLE 7 YOUR TURN 7 Find a polynomial function f such that Find a polynomial function f such that Degree: Degree: its zeros are 4, 1, and and its zeros are, 4, and 1 and f() = 5. f() = 16. We do not know the leading coefficient L; f(x) = L(x + 4)(x 1)(x ) Since f() = 5, L( + 4)( 1)( ) = 5 6L = 5 L = 5 6 Therefore, f(x) = 5 (x + 4)(x 1)(x ) 6 EXAMPLE 8 YOUR TURN 8 Find a polynomial f whose zeros and degree are given, Find a polynomial f whose zeros and degree are given, and the leading coefficient is 1. and the leading coefficient is 1. Zeros: of multiplicity, Zeros: of multiplicity, of multiplicity 1, and 1 of multiplicity, and 1 of multiplicity ; 1 of multiplicity ; degree = 5 degree = 7 Since the sum of multiplicities is 5, we can find a polynomial; f(x) = 1 (x + ) (x )(x 1) = (x + ) (x )(x 1)

160 160 COMPLEX ZEROS: If a polynomial f(x) has positive degree, all real coefficients, and has a complex zero a + bi, then f(x) has also zero a bi. f(x) = (x a bi)(x a + bi)g(x) = (x ax + a + b )g(x) EXAMPLE 9 YOUR TURN 9 Find a polynomial function f of degree such that each Find a polynomial function f of degree such that each coefficient is real, and it has the zeros 1, and 4 + i. coefficient is real, and it has the zeros, and 1 i. There are three zeros Then f(x) zeros i 4 i 1, 4 + i, 4 i = (x + 1) (x + 4 i)(x i) distributive law factors x + 1 x + 4 i x i = (x + 1)(x + 4x + xi 4x i xi 4i i ) = (x + 1)(x + 8x + 17) EXAMPLE 10 YOUR TURN 10 Find a polynomial function f of degree 4 such that each Find a polynomial function f of degree 4 such that each coefficient is real, and it has the zeros, 1, and i. coefficient is real, and it has the zeros, 4, and 1 + i. There are four zeros, 1, + i, i Then f(x) = (x )(x + 1)(x i)(x + i) = (x )(x + 1)(x 4x + 1)

161 161 PRACTICE PROBLEMS 1. Find the leading term and the leading coefficient of the polynomial. (A) f(x) = x 4 + x + x 7 (B) f(x) = 4x 7 + x 6 (C) f(x) = x (x )(x + ) (D) f(x) = (x ) (x 1) (A) x 4 ; 1 (B) 4x 7 : 4 (C) x 5 : (D) 8x 5 : 8. Graph the following function without calculator (A) f(x) = (x + ) (x 4) (B) f(x) = x (x 1) (x ) (C) f(x) = x x + 8x 16 (D) f(x) = x 5 9x. Find the zeros and its multiplicity of the functions. Decide whether it crosses or touches the x-axis at each zero. (A) f(x) = 6(x 4)(x + 5) (B) f(x) = 5x (x 1) (x + ) (C) f(x) = (x x 6) (x ) 4 (D) f(x) = (x x 8) (x 4) 8 (E) f(x) = x 5 1x x (F) f(x) = x x 4x Find a polynomial f(x) whose zeros and degree are given: (A) Zeros: 1 of multiplicity 1, of multiplicity, and of multiplicity ; degree = 5 (B) Zeros: of multiplicity, of multiplicity, and 4 of multiplicity ; degree = 7 (C) Zeros: 1 of multiplicity, of multiplicity 1, and 5 of multiplicity ; degree = 6 (D) Zeros are, 1,, its degree is, and f(1) = 5 (E) Zeros are 4,, 6, its degree is and f() = 8 (F) Zeros are 4 with multiplicity 1, with multiplicity and f() = 5 (A) 4 with mult 1, 5 with mult (B) 0 with mult 1 with mult with mult (C) with mult 7 with mult (D) 4 with mult with mult 10 with mult 8 (E) 0 with mult, with mult (F) with mult, with mult 1 (A) (x + 1)(x ) (x ) (B) (x ) (x + ) (x 4) (C) (x + 1) (x )(x + (D) (E) (F) 5) 5 4 ) (x )(x + 1)(x 1 (x + 4)(x 6 )(x + 6) 5 7 (x + 4)(x ) 5. Find a polynomial f(x) whose zeros and degree are given: (A) Zeros: of multiplicity, 1 i of multiplicity 1; degree = 4 (B) Zeros: of multiplicity 1, i of multiplicity 1; degree = (C) Zeros: i of multiplicity 1, i of multiplicity 1; degree = 4 (D) Zeros: of multiplicity, i of multiplicity 1, and + i of multiplicity 1; degree 6 (E) Zeros:, i; degree ; f() = 4 (A) (x ) (x x + 5) (B) (x )(x + 9) (C) (x 4x + 5)(x + 4) (D) (x ) (x 6x + 10)(x 4x + 1) (E) 4 (x 9 )(x 4x + 1)

162 16 6. Find a polynomial f with the smallest degree whose graph is in the figure. (A) (B) (A) f(x) = 1 (x + ) (x 1)(x 4) (B) f(x) = 1 (x + 5 4)(x ) (-,0) (1,0) (4,0) 5 5 (,-5) (-4,0) (,0) (1,-1) 7. Which of following is a factor of a polynomial x 5x 4x +? (A) (x + ) (B) (x 1) (C) (x 1) (D) (x + 1) C 8. Which of following is a factor of a polynomial x 7x 16x + 1? (A) x + (B) x 1 (C) x + (D) x A 9. Which of following is a factor of a polynomial x + 7x 4? (A) x + 5 (B) x (C) x (D) x C 10. Which of following is a factor of a polynomial x 4 7x x + 1x + 6? (A) x 5 (B) x + 1 (C) x + (D) x + B 11. Which of following is a factor of a polynomial x 7x + x? (A) x (B) x 1 (C) x + (D) x + B 1. Describe the end behavior of the function (A) f(x) = x 4 + x x + 6x (B) f(x) = 5x 6 + 6x 5 x + 8 (C) f(x) = x 7 + 4x 5 + x 1 (D) f(x) = 7x + x x + 9 (A) x, y ; x, y (B) x, y ; x, y (C) x, y ; x, y (D) x, y ; x, y

163 16 LECTURE 4- DIVISION OF POLYNOMIALS Video 1) Synthetic division Method ) Long division Method ) Reminder theorem There are two methods: 1) Synthetic division Method: when divisor is of the form x + a or x a ) Long division Method: anytime you can use. Example 1: Use synthetic division, find Quotient and Remainder when x 5x + x + 7 is divided by x Solution: (We can only use the synthetic method when degree of the divisor is 1) Step 1: To set up the problem, first, set the divisor equal to zero to find the number to put in the left of division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficients of the dividend. 5 7 Step : Bring the leading coefficient (first number) straight down. 5 7 Step : Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step 4: Add the two numbers together and write the result below the line Step 5: Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step 6: Add the two numbers together and write the result below the row

164 164 Step 7: Multiply the number in the division box with the number you brought down and put the result in the next column above the line Step 8: Add the two numbers together and write the result below the row Step 9: Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the other numbers below the line are the coefficients of the quotient in order by the degree of the term. x 5x + x + 7 x Quotient: x x + 1 Remainder: 9 = x x x Example : Use synthetic division, find Quotient and Remainder when x + 5x + 9 is divided by x + Solution: (We can only use the synthetic method when degree of the divisor is 1) Step 1: To set up the problem, first, set the divisor equal to zero to find the number to put in the left of division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficients of the dividend Step : Bring the leading coefficient (first number) straight down Step : Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step 4: Add the two numbers together and write the result below the line Step 5: Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step 6: Add the two numbers together and write the result below the line

165 165 Step 7: Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step 8: Add the two numbers together and write the result below the line Step 9: Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the other numbers below the line are the coefficients of the quotient in order by the degree of the term. x + 5x + 9 x + Quotient: x x + Remainder: 0 = x x + Example : Use synthetic division, find Quotient and Remainder when x 5x + x + 7 is divided by x Solution: (We can only use the synthetic method when degree of the divisor is 1) Step 1: Once the problem is set up correctly, bring the leading coefficient (first number) straight down Step : Multiply the number in the left of the division box with the number you brought down and put the result in the next column above the line Step : Add the two numbers together and write the result below the line Step 4: Repeat steps and 4 until you reach the end of the problem Step 5: Write the final answer. Quotient: x x and Remainder: x 5x + x + 7 x = x x + x

166 166 YOUR TURN Find its Quotient and Remainder if you divide 5x 7x + x 9 by x + YOUR TURN Find its Quotient and Remainder if you divide x 4 + 5x 6x + x + by x YOUR TURN Find its Quotient and Remainder if you divide x 4 + 5x 7x 5 by x + YOUR TURN Find its Quotient and Remainder if you divide x 4 4x + x 6 by x 1

167 167 Example 4: Use long division, find Quotient and Remainder when x x + 6 is divided by (x 1) Solution: Step 1: Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term. Step : Divide the term with the highest power in dividend by the term with the highest power in divisor. x x = x Step : Multiply the divisor and the answer obtained in the previous step. Write it under the dividend, lining up terms of equal degree. x(x + 0x 1) = x + 0x x Step 4: Subtract last line from the line above it. Step 5: Now, repeat the procedure. In this case, x = x x + 0x 1 x x + 0x + 6 x x + 0x 1 x x + 0x + 6 x x + 0x 1 x x + 0x + 6 x + 0x x x x + 0x 1 x x + 0x + 6 x + 0x x x + x + 6 x x + 0x 1 x x + 0x + 6 x + 0x x x + x + 6 Step 6: Multiply the divisor and the answer obtained in the previous step. Write it under the last line, lining up terms of equal degree. (x + 0x 1) = x + 0x + x x + 0x 1 x x + 0x + 6 x + 0x x x + x + 6 x + 0x + Step 7: Subtract last line from the line above it. (Since degree of the resulting polynomial is less than degree of the divisor, we stop here!) x x + 0x 1 x x + 0x + 6 x + 0x x x + x + 6 x + 0x + x + 4 Step 8: Write the final answer. Quotient: x ; Remainder: x + 4 x x + 6 x 1 = x + x + 4 x 1 YOUR TURN Find Quotient and Remainder if you divide x + 4x + x 8 by x + x YOUR TURN Find Quotient and Remainder if you divide x x + 9 by x + 1

168 168 REMAINDER/ FACTOR THEOREM: Let Q be the quotient and R be its remainder when a polynomial f is divided by x c, f(x) = (x c)q + R (A) Remainder: f(c) (B) If the value of f is zero when x = c, then f has a factor x c (f is divisible by x c) EXAMPLE 5 YOUR TURN 5 Use the factor theorem to decide whether x 1 is a factor Use the factor theorem to decide whether x + is a factor of a polynomial f(x) = 4x x + x. of a polynomial f(x) = x x + x. We know that the following are equivalent; (A) (x a) is a factor of f(x) (B) x = a is a x-intercept of y = f(x) (C) f(a) = 0 We only check whether f(1) is zero or not. f(1) = 4(1) (1) + (1) = 0 Therefore, x 1 is a factor of f(x). EXAMPLE 6 YOUR TURN 6 Find the remainder when f(x) = x x + 1 is Find the remainder when f(x) = 5x 99 x + 8 is divided by x + 1. divided by x 1. By Remainder theorem, the remainder is f(1) = ( 1) ( 1) + 1 =

169 169 EXAMPLE 7 YOUR TURN 7 Find the constant k such that f(x) = x kx + kx + 4 Find the constant k such that has a factor x. f(x) = x + kx 5kx + has a factor x. Since f(x) has a factor x, x = is x-intercept of f. f() = 0 () k() + k() + 4 = 0 8 4k + k + 4 = 0 k = 1 k = 6 EXAMPLE 8 YOUR TURN 8 If f(x) = x 4 + ax + 5x 9x 0 is divisible by If f(x) = 6x x + ax + 1 is divisible by x +, (x + 1), find the value of a. find the value of a Since f(x) is divisible by x + 1, x = 1 is a x-intercept of f(x); f( 1) = 0 ( 1) 4 + a( 1) + 5( 1) 9( 1) 0 = 0 55 a = 0 a = 55

170 170 PRACTICE PROBLEMS 1. Find quotient and remainder when we divide (using the synthetic method) (A) 4x 7x 11x + 5 by x 1 (B) x 7 by x (C) x 17x + 15x 5 by x (D) x 4 x + x + 1 by x + 1 (E) 4x + 7x 1x + by x + (F) x 4 x 5 by x 1 (G) x 6x + 11x 6 by x (H) x x 4x + 5 by x 1 (I) x + 5x 4 by x + (J) x 7x + 9 by x (K) x + x 6x + by x 1 (L) x x 1x 7 by x 1 (M) x + x x + 5 by x 1 (N) x + 5x 4x + 1 by x + 1 (O) x 5x + x + 5 by x + 1 (P) 4x x + by x 1. Find quotient and remainder when we divide (using the long division) (A) 4x + 4x + x 5 by (x 1) (B) x 4x + 11x + 8 by (x + ) (C) x + x 10 8x by (x + 1) (D) x 4 x + x 7 by (x + ) (E) x + x + x 5 by (x + x + ) (F) x 5x + 7 by (x x + ) (G) x 1x + 0x 1 by (x x + 1) (H) 4x 4 x 7x 7x by (x x ) (I) 4x 4 + x + 4x + 9x 6 by (x + ) (J) x 4 + 4x + 7x + 5x by (x + x + ). Find the remainder when we divide (A) 5x 7 x 4 + x 6x by x + 1 (B) x 54 x + 5 by x 1 (C) x 67 x + x 5 by x + 1 (D) 5(x ) 15 + x + 1 by x 1 (A) Q: 4x x 14, R: 9 (B) Q: x + x + 9, R: 0 (C) Q: x 11x 7, R: 9 (D) Q: x 5x + 5x, R: (E) Q: 4x 5x +, R: 6 (F) Q: x + x x 1, R: 6 (G) Q: x 4x +, R: 0 (H) Q: x x 5, R: 0 (I) Q: x x +, R: 1 (J) Q: x + x +, R: 15 (K) Q: x + 4x 4, R: 1 (L) Q: x 1, R: 11 (M) Q: x + x 1, R: 4 (N) Q: x + x, R: 4 (O) Q; x x + 1, R: 4 (P) Q: x + x 1, R: 1 (A) Q; x + x +, R: (B) Q: x x + 5, R: (C) Q; x 8, R = x (D) Q: x x, R: 8x (E) Q: x + 1, R; 7 (F) Q: x +, R: x + 1 (G) Q: x 6, R: 5 (H) Q: 4x + 11x + 4 R: 117x + 66 (I) Q: 4x + x 8, (J) R: 18 Q: x + x, R: x (A) 0 (B) 5 (C) 7 (D) 1 4. Is x 1 a factor of a polynomial 17x 18x + 5x 4? Yes

171 Find the values of a and b such that a =, b = 11 (x + a)(x + 5) = 6x + bx 10 for all x 6. Find the constant a such that the following statement is an identity: x x 16 x 4 7. Find the values of a and b such that 4x + 6x + 5 x a 8. Find constants a and b satisfies: = x + + a, for all x except 4 x 4 = x + + b x a for all x except a (A) When x 4 x x x is divided by x x + a, its remainder is x + b a = 8 a = 1, b = 7 (A) a =, b = 4 (B) a = 5, b = 7 (C) a = 5, b = 4 (B) When x + ax + b is divided by x x +, its remainder is x + 1. (C) When x + ax + bx + is divided by x 1, its remainder is 5x 9. Find the constants a, b, c and d if (A) x + ax a x + 10 is divisible by x 1 (B) ax x + x 6a is divisible by x (C) x 1 is a factor of f(x) = ax 4 ax + x a (D) x x + ax + b is divisible by x + x (A) a = 4, (B) a = 4 (C) a = 1 (D) a = 6, b = 8 (E) a =, b = 1 (F) a =, b = (G) a = 1, b = 4, c =, d = (H) a =, b = 18, c = 17, d = 4 (E) x + ax + b is divisible by x + x 1 (F) x ax + = (x 1) (x + b) (G) x x x + 6 = a(x 1) + b(x 1) + c(x 1) + d (H) x x + 5x + 6 = a(x + 1) + b(x + 1) + c(x + 1) + d 10. Find the remainder if 4(x + 1) is divided by x Evaluate x 4 x 6x + x + 5 if x x 1 = Find the remainder when we divide 1001 by 7

172 17 LECTURE 4- ZEROS OF POLYNOMIALS Video 1) Rational zero theorem ) Find all zeros ) Find all complex/real zeros (1) 4) Find all complex/real zeros () NOTE: Let f(x) be a polynomial with real coefficient. x-intercepts are the real zeros of f(x) f(x) = x x 4 f(x) = x 6x + 10 Real/complex zeros 0 = x x 4 x = 1, 4 : real 0 = x 6x + 10 x = ± i 4 4 The graph of y = f(x) 5 5 y = x x y = x 6x Two real x intercepts No x intercepts (two complex zeros) The number of all real/complex zeros of f(x) is the degree of f(x). RATIONAL ZERO (OR ROOT) THEOREM: If f(x) = a n x n + a n 1 x n a 1 x + a 0, where a polynomial where a j s are integer coefficient and a n 0 and the reduced fraction p is a rational zero, then p is a factor of the q constant term a 0 and q is a factor of the leading coefficient a n. is a solution of x + 7x + 6 = 0 ; is a factor of, is a factor of 6

173 17 EXAMPLE 1 YOUR TURN 1 List all possible rational zeros of the polynomial List all possible rational zeros of the polynomial f(x) = x 5x 4 f(x) = 15x + 14x x Find all possible rational solutions factor of constant term (± factor of leading coefficient ) Factors Constant: 4 1,, 4 Leading coefficient: 1, All possible rational solutions are ± 1 1, ± 1, ± 1, ±, ± 4 1, ± 4 EXAMPLE YOUR TURN Let f(x) = x 7 + ax bx 9 be a polynomial with Let f(x) = 5x 5 + ax bx + 4 be a polynomial with integer coefficients (i.e., a, b are integers). Which one of integer coefficients (i.e., a, b are integers). Which one of following is definitely not a possible zero of f(x)? following is definitely not a possible zero of f(x)? (A) 1 (B) 9 (C) (A) 1 5 (B) 4 5 (C) (D) (E) 1 (D) 5 (E) 1 By the rational zero theorem, all possible rational zeros are; ± 1 1, ± 1, ± 1, ±, ± 9 1, ± 9 Factors Constant: 9 1,, 9 Leading coefficient: 1, Then, cannot be a rational solution of f.

174 174 EXAMPLE YOUR TURN Find all real zeros of the polynomial and write f in the Find all real zeros of the polynomial and write f in the product of factors. product of factors. f(x) = x x 5x + 6 f(x) = x x x + Step 1: Find all possible rational solutions Factors Constant: 6 1,,, 6, Leading coefficient: 1 1 All possible rational solutions are ± 1 1, ± 1, ± 1, ± 6 1 Step : Find the rational solutions among the above values by using your calculator. (three x-intercepts means three real zeros) y = x x 5 x f( ) = f(1) = f() = 0 Zeros are, 1, and Step : Since the leading coefficient is 1 and zeros are,1,; f(x) = (x + )(x 1)(x )

175 175 EXAMPLE 4 YOUR TURN 4 Find all real zeros of the polynomial and write f in the product of factors. f(x) = 6x 11x 4x + 4 Find all real zeros of the polynomial and write f in the product of factors. f(x) = 6x + 11x 4x 9 Step 1: First, find all possible rational zeros: Factors Constant: 4 1,, 4 Leading coefficient: 6 1,,, 6 All possible rational zeros are ±1, ±, ±4, ± 1, ± 1, ±, ± 4, ± 1 6 Step : Find the rational zeros among the above values by using your calculator. (three x-intercepts means three real zeros) Since this polynomial has degree, all zeris are, 1, Step : Find the factor form By leading coefficient 6 and step, f(x) = 6(x ) (x 1 ) (x + ) = (x )(x 1)(x + )

176 176 EXAMPLE 5 YOUR TURN 5 Find all real zeros of the polynomial and write f in the Find all real zeros of the polynomial and write f in the product of factors. product of factors. f(x) = 7x x + 7x 1 f(x) = x x + x Step 1: First, find all possible rational zeros: Factors Constant: 1 1 Leading coefficient: 7 1, 7 All possible rational zeros are ±1, ± 1 7 Step : Find the rational zeros among the above values by using your calculator (one x-intercepts means one real zero) y = 7 x x + 7 x 1 15 This is only one real solution 1 7 Step : Find the quotient when f(x) is divided by x 1 7 1/ The quotient is 7x + 7 = 7(x + 1) f(x) = 7(x + 1) (x 1 7 ) = (x + 1)(7x 1)

177 177 EXAMPLE 6 YOUR TURN 6 Find all real zeros of the polynomial and write f in the Find all real zeros of the polynomial and write f in the product of factors. product of factors. f(x) = x x 7x + 14 f(x) = x + x 0x 0 Step 1: First, find all possible rational zeros: Factors Constant: 14 1,, 7, 14 Leading coefficient: 1 1 All possible rational zeros are ±1, ±, ±7, ±14 Step : Find the rational zeros among the above values by using your calculator (three x-intercepts means three real zeros) y = x 5 x 7 x This is only one rational solution: Step : Find the quotient when f(x) is divided by x The quotient is x 7. The other two real solutions satisfy x 7 = 0. x = 7 x = ± 7 f(x) = (x )(x 7)(x + 7)

178 178 YOUR TURN Find all real zeros of a polynomial f and write f in the product of factors. f(x) = x + x 7x + YOUR TURN Find all real zeros of a polynomial f and write f in the product of factors. f(x) = x 4 + 7x + x 7x YOUR TURN Find all real zeros of a polynomial f and write f in the product of factors. f(x) = x 4 + 4x 19x 8x + 1 YOUR TURN Find all real zeros of a polynomial f and write f in the product of factors. f(x) = 15x 9x 8x + 1

179 179 EXAMPLE 7 YOUR TURN 7 Find the real/complex zeros of a polynomial Find the real/complex zeros of a polynomial f(x) = x 4 x + 11x 7x + 18 f(x) = x 4 + x x + 4x 4 Step 1: First, find all possible rational zeros Factors Constant: 18 1,,, 6, 9, 18 Leading coefficient: 1 1 All possible rational zeros are ±1, ±, ±, ±6, ±9, ±18 Step : Find the rational zeros among the above values by using your calculator. 5 y = x 4 x + 11 x 7 x + 18 Therefore {1, } are only rational zeros. Step : Use synthetic method twice by Step to find quotient Therefore: the quotient is x + 9; Step 4: To find the other solutions which satisfy x + 9 = 0 x = 9 x = ± 9 = ±i Step 5: The zero set is {1,, i, i}

180 180 YOUR TURN Find the real/complex zeros of a polynomial f(x) = x 4 5x + x + 19x 0 YOUR TURN Find the real/complex zeros of a polynomial f(x) = x 4 + x 6x 18x + 0

181 181 EXAMPLE 8 YOUR TURN 8 Solve the equation: Solve the equation: 6x + 7x 16x 1 = 0 15x 6x 1x + 8 = 0 Step 1: First, find all possible rational solutions Factors Constant: 1 1,,, 4, 6, 1 Leading coefficient: 6 1,,, 6 All possible rational solutions are ±1, ±, ±, ±4, ±6, ±1 ± 1, ±1, ±, ±, ±, ±6 ± 1, ±, ±1, ± 4, ±, ±4 ± 1 6, ± 1, ± 1, ±, ±1, ± Step : Find the rational solutions among the above values by using your calculator x =,, f( x) = 6 x + 7 x 16 x

182 18 PRACTICE PROBLEMS 1. List all possible rational zeros of (A) f(x) = x + x 8x + 5 (B) f(x) = x 9x + 1x 15 (C) f(x) = x 4 x + 14x x 0 (D) f(x) = x 4 + 5x x + 7x + 14 (A) ±1, ±5, ± 1. ± 5 (B) ±1, ±, ±5, ±15 (C) ±1, ±, ±, ±5, ±10, ±15, ±0 (D) ±1, ±, ±7, ±14. Find all real solutions of the equation (A) x + x 8x + = 0 (B) x 4 x + 6x + x 60 = 0 (C) 6x 4x + x = 0 (D) x 4 5x + 4x + 10x 1 = 0 (A) 1, 1, (B), (C) (D),, ±. Find the real zeros of the polynomial and write the zeros to factor completely (A) f(x) = x 5x 4x + (B) f(x) = x + x 1x 4 (A) 1,, 1: f(x) = (x 1)(x )(x + 1) (C) f(x) = x x + 6x 9 (D) f(x) = 6x x + 5x 6 (B), 1, ; f(x) = (C) (x + )(x + 1)(x ) ; f(x) = (x )(x + ) (D),, 1 ; f(x) = (x 4. Find the real/complex zeros of the polynomial (A) f(x) = x 5x (B) f(x) = x x 5x + 15 (C) f(x) = x 4 x 5x + x + 6 (D) f(x) = x 4 5x x + 5x + 6 (E) f(x) = x 4 x 8x 40 (F) f(x) = x 4 + x x + 4x 4 (G) f(x) = x 4 x + 7x + 1x 6 (H) f(x) = x 4 + 6x + 10x + 6x + 9 )(x )(x 1) (A),1 ± (B), ± 5 (C) 1,, ± (D),, ± 5 (E) 1,4, 1 ± i (F),, ±i (G),1, ± i (H), ±i (I) 4,5, ±i (J),4, ± 5 (I) f(x) = x 4 x 17x x 60 (J) f(x) = x 4 x 17x + 5x + 60 (K) f(x) = x 7x + 19x 1 (L) f(x) = x 11x + x 15 (K), ± i (L), ± i (M),1 ± i (N) 1, ± i (M) f(x) = x 4 + x + x + 1x + 0 (N) f(x) = x 4 8x + x 6x + 10

183 18 LECTURE 4.4 RATIONAL FUNCTION Video 1) Introduction to rational function ) Find Vertical/Horizontal/Oblique Asymptote DOMAIN OF RATIONAL FUNCTION: If P(x) and Q(x) are polynomials, then a function of the form f(x) = P(x) Q(x) is called a rational function, if Q(x) is not the zero polynomial. (A) Zero(s) of (x) : all x-values c such that P(c) = 0 and Q(c) 0. (B) Domain of (x) : all real number except x-values c such that Q(c) = 0. ASYMPTOTE (A) Vertical Asymptote; The line x = a is a vertical asymptote for the graph of a function if f(x) or f(x) as x approaches a from either the left or the right. (B) Horizontal Asymptote: The line y = c is a horizontal asymptote for the graph of function f if f(x) c as x or x horizontal Asymptote Vertical Asymptote HOW TO FIND VERTICAL/HORIZONTAL/OBLIQUE ASYMTPTOTES? Let f(x) = P(x) Q(x) be a rational function. (A) Vertical Asymptote; line x = c where P(c) 0 and Q(c) = 0. (B) Horizontal Asymptote 1) If deg P(x) > deg Q(x), there are no Horizontal Asymptote. ) If deg P(x) = deg Q(x), Horizontal Asymptote is Line y = Leading coefficient P(x) Leading coefficient Q(x) ) If deg P(x) < deg Q(x), line y = 0 is Horizontal Asymptote. (C) Oblique Asymptote; only when deg P(x) = deg Q(x)+1, oblique asymptote is y = quotient when P(x) is divided by P(x). oblique asymptote oblique asymptote

184 184 EXAMPLE 1 YOUR TURN 1 Consider f(x) = x +x 6 x x (A) Find the domain of f. (B) Find the hole if any (C) Find x-intercept (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function. Consider f(x) = x 9 x x 8 (A) Find the domain of f. (B) Find the hole if any (C) Find x-intercept (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function. 1) First, find the graph of the function; One-V. A One- H.A None- O. A ) Factor the numerator and denominator and find the x values which make each zero. f(x) = x + x 6 (x + )(x ) x = x (x + 1)(x ), 1, - Domain: all real numbers except 1, - Hole: x value(s) in which both sides are zero: - x-intercept: only make numerator = 0: - Vertical Asymptote(s): it connects with x values in which only denominator is zero; Vertical line x = 1 - Horizontal Asymptote: both sides has same degree; y = leading coefficient of numerator leading coefficient of denominator Horizontal line y = 1 - Oblique Asymptote: None

185 185 EXAMPLE YOUR TURN Consider f(x) = x+ x x 10 (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function. Consider f(x) = x+ x x 6 (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function 1) First, find the graph of the function; two V.A One H.A No O. A ) Factor the numerator and denominator and find the x values which make each zero. f(x) = x + x x = x + (x + )(x 5), 5 - Domain: all real numbers except,5 - Hole: x value in which both sides are zero: None - x-intercept: only make numerator = 0: - Vertical Asymptote(s): it connects with x values in which only denominator is zero; Vertical line x =, x = 5 - Horizontal Asymptote: numerator has 1 less degree; Horizontal line y = 0 - Oblique Asymptote: None

186 186 EXAMPLE YOUR TURN Consider f(x) = x +4x+ x (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function Consider f(x) = x 4 x+1 (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function 1) First, find the graph of the function; one V.A no H.A one O. A ) Factor the numerator and denominator and find the x values which make each zero. f(x) = (x + )(x + 1) x, 1 - Domain: all real numbers except - Hole: x value in which both sides are zero: None - x-intercept: only make numerator = 0:, 1 - Vertical Asymptote(s): it connects with x values in which only denominator is zero; Vertical line x = - Horizontal Asymptote: None - Oblique Asymptote: find the quotient: x + 6 y = x + 6

187 187 EXAMPLE 4 YOUR TURN 4 Consider f(x) = x x +4 (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function Consider f(x) = x4 +1 x (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function 1) First, find the graph of the function; x y = x + 4 no V.A one H.A no O. A ) Factor the numerator and denominator and find the x values which make each zero. f(x) = x x none - Domain: all real numbers - Hole: x value in which both sides are zero: None - x-intercept: only make numerator = 0: 0 - Vertical Asymptote(s): it connects with x values in which only denominator is zero; None - Horizontal Asymptote: numerator has 1 less degree; Horizontal line y = 0 - Oblique Asymptote: find the quotient: None

188 188 YOUR TURN Consider f(x) = (x x 6)(x+4) (x x )(x 16) (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function YOUR TURN Consider f(x) = (x +x)(x 1) (x x+)(x 1) (A) Find the domain of f. (B) Find the hole(s) if any (C) Find x-intercept(s) (D) Find vertical asymptote(s) of this function. (E) Find the horizontal asymptote(s) of this function. (F) Find the oblique asymptote(s) of this function

189 189 EXAMPLE 5 YOUR TURN 5 Find an equation of a rational function R(x) that 1) Vertical Asymptote: x = 5, x = ) Horizontal Asymptote: y = 1 ) x-intercept:, 5 Find an equation of a rational function R(x) that 1) Vertical Asymptote: x =, x = 4 ) Horizontal Asymptote: y = ) x-intercept:, 5 By 1) R(x) = (x + 5)(x ) By ) By ) R(x) = R(x) = A(x )(x 5) (x + 5)(x ) ( 1)(x )(x 5) (x + 5)(x ) EXAMPLE 6 YOUR TURN 6 Find an equation of a rational function R(x) that Find an equation of a rational function R(x) that We know that 1) Vertical Asymptote: x =, x = 4 ) Horizontal Asymptote: y = 1 ) x-intercept: 0 with multiplicity even R(x) = 1 x (x + )(x ) = x x 4

190 190 PRACTICE PROBLEMS Consider the following function to find the following. (A) Find the domain. (B) Find the reduced form (C) Find hole if any (D) Find intercepts (E) What is the equation of the vertical asymptote(s) of this function? (F) What is the equation of the horizontal asymptote(s) of this function? (G) What is the equation of the oblique asymptote(s) of this function? 1 f(x) = x x x 1 f(x) = x x +4x 1 f(x) = x x 6 x x 0 4 f(x) = x x 6 x +x 18 5 f(x) = x x+1 6 f(x) = x x 6 x 7 f(x) = x +x 1 x+ 10 f(x) = x4 +x + x+1 8 f(x) = x +x 8 x 11 f(x) = x+ x +1 9 f(x) = x x 4x+ 1 Find an equation of a rational function R(x) such that (A) Vertical Asymptote: x = Horizontal Asymptote: y = 1 x-intercept: (B) Vertical Asymptote: x =, x = Horizontal Asymptote: y = x-intercept: 4 (C) Vertical Asymptote: x =, x = 1 Horizontal Asymptote: y = 0 x-intercept: R() = 1 (D) Vertical Asymptote: x = Horizontal Asymptote: y = 1 x-intercept: 5 Hole at x = (E) 4 (F)

191 191 SOLUTIONS 1. (A)(, ) (,4) (4, ) (B). (A)(, 6) ( 6,) (, ) (B) 1 x x x 1 x+6 (C)None (D) (,0), (0, 1 ) (E)x = 4, x = (F) y = 0 (G)None 6 (C)x = (D) None (E) x = 6 (F) y = 0 (G)None. (A)(, 5) ( 5,6) (6, ) (B) x x 6 (C)None (D) (,0), (,0), (0, 1 ) (E)x = 5, x = 6 (F) y = 1 (G)None x x (A)(, 6) ( 6,) (, ) (B) x+ x+6 (C)x = (D) (,0), (0, 1 ) (E) x = 6 (F) y = 1 (G)None 5. (A)(, 1) ( 1, ) (B) x (C)None (D) x+1 (, 0), (0, ) (E)x = 1 (F) y = (G)None 6. (A)(, ) (, ) (B)x + (C)x = (D) (,0), (0,) (E)None (F)None (G) y = x + 7. (A)(, ) (, ) (B) x +x 1 x+ 8. (A)(, ) (, ) (B) x +x 8 x 9. (A)(, 1) (1,) (, ) (B) 1 (C)None (D)( 4,0), (,0), (0, 6) (E)x = (F)None (G) y = x 1 (C)None (D)( 4,0), (,0) (0, 8 ) (E) x = (F)None (G) y = x + 5 x (A)(, 1) ( 1, ) (B) x4 +x + x (A)(, ) (B) x+ x (A)y = x x (B)y = (C)x = (D) (0, 1) (E) x = 1 (F) y = 0 (G)None (C)None (C)None (D)(,0), (0,) (E)None (x 4) (x )(x+) (C)y = 10(x ) (x+)(x 1) (D) (0,) (E) x = 1 (F)None (G)None (D)y = (x 5)(x+) (x )(x+) (F)y = 0 (G)None (E)y = x 4 (x+1)(x ) (F)y = (x 1) (x+)(x )

192 19 LECTURE 4-5 INEQUALITIES Video 1) Quadratic inequalities (1) ) Quadratic inequalities () ) Polynomial inequalities 4) Rational inequalities EXAMPLE 1 YOUR TURN 1 Solve the inequality x x + 8 Solve the inequality x 6 x 1) Make one side zero x x 8 positive,zero = (x 4)(x + ) 0 ) Find x values which x x 8 = 0 x = 4, ) Decide the sign of f(x) = x x 8 : x 4 Test value f( ) f(0) f(5) x x ) The solution set (positive or zero) is (, ] [4, ) EXAMPLE YOUR TURN Solve the inequality x x 1 Solve the inequality x 4 1) Make one side zero x x 1 = (x 4)(x + ) 0 negative e,zero ) Find x values which x x 1 = 0 x = 4, ) Decide the sign of f(x) = x x 1 : x 4 Test value f( 4) f(0) f(5) x x ) The solution set (negative or zero) is [,4]

193 19 EXAMPLE YOUR TURN Solve the inequality x x < Solve the inequality x x > 10 1) Make one side zero x x = (x )(x + 1) < 0 negative ) Find x values which x x = 0 x =, 1 ) Decide the sign of f(x) = x x x 1 Test value f( ) f(0) f() x x ) The solution set (only negative) is ( 1,) EXAMPLE 4 YOUR TURN 4 Solve the inequality x + x Solve the inequality x 4 1) Make one side zero x + x + 5 positive,zero = (x + 1) ) Find x values which x + x + 5 = 0 No real values ) Decide the sign of f(x) = x + x + 5 x Test value f(0) x + x ) The solution set (only positive) is (, )

194 194 EXAMPLE 5 YOUR TURN 5 Solve the inequality x x + 10 < 0 Solve the inequality x + 5 < 0 1) Make one side zero x x + 10 positive,zero = (x 1) + 9 < 0 ) Find x values which x x + 10 = 0 No real values ) Decide the sign of f(x) = x x + 10 x Test value f(0) x x ) Therefore, there are no solutions or The solution set is EXAMPLE 6 YOUR TURN 6 Solve the inequality: x 1x Solve the inequality x 4x 1) Make one side zero x 1x positive,zero = x (x 4) 0 ) Find x values which x (x 4) = 0 x = 0, 4 ) Decide the sign of f(x) = x 1x : x 0 4 Test value f( 1) f(1) f(5) x 1x ) The solution set (positive or zero) is {0} [4, )

195 195 EXAMPLE 7 YOUR TURN 7 Solve the inequality: x x 9x Solve the inequality: x x 5x + 50 < 0 1) Make one side zero x x 9x + 9 = x (x 1) 9(x 1) 0 positive,zero (x 1)(x 9) = (x 1)(x )(x + ) 0 ) Find x values which (x 1)(x )(x + ) = 0 x =, 1, ) Decide the sign of y = x x 9x + 9 : positive, zero x 1 Test f( 4) f(0) f() f(4) value y ) The solution set (positive or zero) is [,1] [, ) EXAMPLE 8 YOUR TURN 8 Solve the inequality: x x+4 0 Solve the inequality: x+ x 0 1) Make one side zero x x + 4 negative, zero 0 ) Find x values such that Numerator = 0, denominator = 0. x x ) Decide the sign of y = x : negative, zero x+4 x 4 Test f( 5) f(0) f(5) value y + Error 0 + 4) The solution set (negative or zero) is ( 4,]

196 196 EXAMPLE 9 YOUR TURN 9 Solve the inequality: x 1 x x 6 0 1) Make one side zero x 1 x 0 x 6 positive,zero ) Find x values such that numerator = 0 and denominator = 0 x 1 1 (x )(x + ), ) Decide the sign of y = x 1 x x 6 : x 1 Test f( 5) f(0) f() f(5) value y E + 0 E + Solve the inequality: x+1 x x 8 0 4) The solution set (positive or zero) is (,1] (, ) YOUR TURN Solve the inequality: x x +x 6 0 YOUR TURN Solve the inequality: x (x ) (x+)(x 4) 0

197 197 EXAMPLE 10 YOUR TURN 10 Solve the inequality: x+ x 5 1) Make one side zero x + x 5 0 x + 5(x ) x x 0 x + 5(x ) x = x x positve, zero ) Find x values such that numerator = 0 and denominator = 0 x + 18 x ) Decide the sign of y = x+18 x 6 x 6 Test f(0) f(4) f(7) value y Error + 0 4) The solution set (positive or zero) is (,6] : Solve the inequality: 4 x 1 EXAMPLE 11 YOUR TURN 11 Solve the inequality: 1 > 1 x x+ Solve the inequality: 1 x < x 9

198 198 PRACTICE PROBLEMS 1. Solve the inequalities (A) x + x < 6 (B) x(x ) 4 (C) 16 x 0 (D) x(x + 6) 16 (E) x x < 6 (F) 1 + 4x x (G) x 4x (H) x x + 0 (I) x 0 (J) x x + 0 (K) x 4x (L) x + x + > 0 (M) x 5x + > 0 (N) 6x + x < 0 (O) 8x 10x 0 (P) x + x + 1 > 0 (Q) x x + 4 < 0 (R) x + x < 0 (A) (,) (B) (, 1] [4, ) (C) [ 4,4] (D) [ 8,] (E) (, ) (F) [,6] (G) (, ) (H) (I) [, ] (J) [, 1] (K) {} (L) (, ) (M) [, 1] (N) (, ] [, ) (O) [ 1 4, ] (P) (, ) (Q) ( 1,1). Solve the inequalities: (A) (x + )( x)(x + 5) 0 (B) x(x + )(x 4) 0 (C) x 9x (D) x < 4x (E) x x 1x 0 (F) x 7x < 8x (G) x x 5x + 6 > 0 (H) x x + x < 0. Solve the inequalities: (A) x 4 x+ 0 (C) x+ x 6 > 0 (E) x+ x+1 < (G) (I) x+4 (B) 0 x+ x 5 (D) < 0 x+ x+ (F) 5 x (x ) 0 (H) 1 < 1 (x )(x+4) x x+ x x 5 x 1 (K) x 1 x +1 > 0 (M) x 16 (x 1) 0 (J) (L) x x x 10 0 x x 4 0 (N) x 8 x x (A) (, 5] [,] (B) (, ] [0,4] (C) [,0] [, ) (D) (, 0) (0,) (E) [,0] [4, ) (F) (, 1] [0,8] (G) (,1) (, ) (H) (, ) (A) (,4] (B) (, 4] (, ) (C) (, ) (6, ) (D) (,5) (E) (, 1) (0, ) (F) (,6] (G) ( 4,] (, ) (H) (,) (I) (1, 5 ) [,5] (J) (, ) [,5) (K) (1, ) (L) [,] (M) [ 4,1) (1,4] (N) (, ] (0,4]

199 199 MATH 1111 COLLEGE ALGEBRA REVIEW FOR EXAM 4 Work these specifically. Try them WITHOUT your notes or text. 1. Which of the following is a factor of a polynomial f(x) = x 4 + 6x + 9x 4x 1? (A) x + (B) x + 1 (C) x (D) x. Find the zeros and its multiplicity of the following functions. Decide whether it crosses or touches the x-axis at each zero. f(x) = x (x + ) (x 5). Find a polynomial whose zeros and degree are given, and the leading coefficient is 1. Zeros: of multiplicity, 1 of multiplicity, and 1 of multiplicity ; degree = 7 4. Find a polynomial f of degree 5, whose coefficient are real, has the zeros, i and 1 i. 5. Use synthetic division, find Quotient and Remainder if you divide x 4 4x + x 6 by x + 6. Find the remainder when x x + 7 is divided by x Let f(x) = x 7 + ax bx 9 be a polynomial with integer coefficients (i.e., a, b are integers). Which of the following is not a possible zero of f(x)? (A) 1 (B) 9 (C) (D) (E) 1 8. Solve the equation. 6x 9x 7x + 10 = 0 9. Find all real zeros of a polynomial f(x) = 4x + 7x x 10. Find all real zeros and factor a polynomial using the zeros. f(x) = x 4 + 7x + x 7x 11. Find all real/complex zeros of a polynomial: f(x) = x 4 x 8x 40

200 00 1. Find the domain, hole(s), x-intercept(s), asymptotes of the rational function f(x) = x 5x 6 x x 0 1. Find the domain, hole(s), x-intercept(s), asymptotes of f(x) = x x 8 x 14. Find the domain, hole(s), x-intercept(s), asymptotes of f(x) = x x x Find a polynomial which satisfies the following conditions: Vertical Asymptote: x =, x = Horizontal Asymptote: y = x-intercept: 4 with multiplicity 16. Solve the inequality: x(x ) Solve the inequality: x < 9x 18. Solve the inequality: x 4 x Solve the inequality: x+5 x+1 0. Solve the inequality: (x ) (x 5)(x+) < 0

201 01 SOLUTIONS 1. A since is x-intercept.. Zeros are 0 with multiplicity : touch x-axis with multiplicity : touch x-axis 5 with multiplicity : cross x-axis. (x ) (x + 1) (x 1) 4. (x )(x i)(x + i)(x 1 + i)(x 1 i) = (x )(x + 9)(x x + 5) 5. R: 8, Q: x x R = f( 1) = ( 1) ( 1) + 7 = 5 7. ±1, ± 1, ±, ±, ±9, ± 9 : (C) 8., 1, 5 since Constant: 10 1,,5,10 Leading coefficient:6 1,,,6 9. 4x + 7x x = x(4x + 7x ) x = 0, 1 4, = x(4x 1)(x + ) 10. 1,, 1,1 using calculator f(x) = (x + 1)(x + )(x + 1)(x 1) 11. By using calculator, we find two real zeros: 1,4 By twice synthetic method, x + x + 10 = 0 Then zeros are 1,4, 1 ± i 1. Domain: (, 5) ( 5,6) (6, ) Hole at x = 6 x-intercept: ( 1,0) Vertical Asymptote: x = 5 Horizontal Asymptote: y = 1 Oblique Asymptote: None 1. Domain: (, ) (, ) Hole: None x-intercept: (4,0), (,0) Vertical Asymptote: x = Horizontal Asymptote: None Oblique Asymptote: y = x Domain: (, ) (,4) (4, ) Hole: None x-intercept: (,0) Vertical Asymptote: x =, x = 4 Horizontal Asymptote: y = 0 Oblique Asymptote: None

202 0 15. y = (x 4) (x )(x+) 16. (, 1] [4, ) x x 4 positve,zero = (x 4)(x + 1) 0 x 1 4 x x (, ) (0,) Let y = x(x )(x + ) = x 9x negative < 0 x 0 y (,4] x 4 x + negative,zero 0 x 4 x 4 x + + E (, 1) [, ) x+5 = x+5 (x+1) = x+1 x+1 x+ x+1 negative,zero 0 x 1 x + x + 1 E (,) (,5) y = (x ) < 0 (x 5)(x + ) negative x 5 y + E 0 E +

203 0 MORE PRACTICE PROBLEMS FOR TEST 4 1. Form a polynomial of degree 6 with zeros: of multiplicity, 1 of multiplicity, and of multiplicity 1. f(x) = (x + ) (x 1) (x ). Find a polynomial function with real coefficients that Degree 4; zeros:, 1, i f(x) = (x )(x + 1) (x 6x + 1). List all the potential rational zeros of P(x) = x 4 + ax + bx 18x + where a, b are real numbers. ±1, ±, ± 1, ± 4. Find a polynomial in the figure whose degree is minimal; f(x) = x(x + )(x ) 5. Find the quotient and remainder (A) Divide x 17x + 15x 5 by x 5 (B) Divide x 4 7x 11x + 5 by x 1. (A) Q: x x + 5, R: 0 (B) Q: x + x 4x 15, R: 10 (C) Q; x 8, R = x (C) Divide x 8x + x 10 by (x + 1) 6. Find all real/complex zeros of a polynomial (A) f(x) = x 7x + x (B) f(x) = x 4 + 4x 19x 8x + 1 (C) f(x) = x 4 + x 14x x + 4 (A) 0,, 1 (B),, 1, (C) ±, 4, (D),, ± i (E),1,1 ± i (D) f(x) = x 4 + 6x + 6x 4x 40 (E) f(x) = x 4 x + 16x Find all zeros of a polynomial and write the real zeros to factor f(x) completely (A) f(x) = x 4 + 5x 17x 1x + 6 (B) f(x) = 5x 4 + x 5x + x + 6 (A), 1, 1, ; f(x) = (x + )(x + 1) (x 1)(x ) (B), 1, 1, 1; f(x) = (x + )(x 1) (x )(5x + 1) 8. Solve the equation: 6x x + 5x 6 = 0,, 1 9. Find the remainder when f(x) = x x + 7 is divided by x Which of the following is a factor of a polynomial f(x) = x 4 + 7x 6x 49x + 0? (A) x 5 (B) x + 1 (C) x + (D) x + C

204 Which of the following is a factor of a polynomial f(x) = x + 5x 8x 15? (A) x 4 (B) x (C) x 5 (D) x + 1 B 1. Solve the inequalities (A) x x 6 (B) x x > 1 (C) x x < 0 (D) x + x + 4 < 0 (E) x x (F) x x 0x 0 (G) x+ x 0 (I) (x 4) (x+)(x 6) 0 x 5 (H) 0 x+ (J) 4x x+ 1. Using the following rational function, answer the question. (1) Find the domain of g(x). () Find the x-intercept(s) of g(x) if any () Find the y-intercept(s) of g(x) if any (4) Find the hole(s) if any (5) Find the vertical asymptote of g(x) if any. (6) Find the horizontal asymptote of g(x) if any. (7) Find the oblique asymptote of g(x) if any. (A) f(x) = x 6x 7 x+ (C) f(x) = x+ x +x 8 (B) f(x) = x x 1 x +x 6 (D) f(x) = x 7x+ x (A) [,] (B) (, ) (4, ) (C) ( 4,5) (D) (E) (, 4] [0,4] (F) [ 4,0] [5, ) (G) [,) (H) (,5] (I) (,4] (6, ) (J) (, ) [6, ) (A) (1)(, ) (, ) ()9, () 9 (4)None (5)None (6)None (7)y = x 9 (B) (1)(, ) (,) (, ) () () (4) (5)x = (6)y = 1 (7)None (C) (1)(, 4) ( 4,) (, ) () () 1 4 (4)None (5)x =, x = 4 (6)y = 0(7)None (D) (1)(, ) (, ) (), 1 () (4)None (5)x = (6)None (7)y = x 9 (E) (1)(, ) (,) (, ) (E) f(x) = x +x 15 x x 6 (F) f(x) = x 4 x 4 (F) () 5 () 5 (4) (5)x = (6)y = 1(7)None (1)(, ) (,) (, ) ()4 ()1 (4)None (5)x =, x = (6)y = 0 (7)None

205 05 UNIT 5. EXPONENTIAL AND LOGARITHMIC FUNCTIONS LECTURE 5-1 COMPOSITION OF FUNCTIONS DEFINITION: The composition of the function g with f is denoted by g f and is defined by the equation (g f)(x) = g(f(x)) The domain of the composite function of g f is the set of all x such that x is in the domain of f and f(x) is in the domain of g. EXAMPLE 1 YOUR TURN 1 Use the table to answer the questions x f(x) g(x) Use the table to answer the questions x f(x) g(x) (A) (f g)( ) = f(g( )) = f( 1) = (A) (f g)() (B) (g f)(1) x f(x) g(x) (B) (f g)() = f(g()) = f() = 1 x f(x) g(x)

206 06 EXAMPLE YOUR TURN Use the graphs in the figure to answer the questions Use the graphs in the figure to answer the questions (A) (f g)( ) (B) (g f)(1) (g f)(1) = g(f(1)) = g( 1) = 1 EXAMPLE YOUR TURN Let f(x) = x and g(x) = x x + 5. Find Let f(x) = x and g(x) = x x 4. Find (A) (f g)(x) (B) (g f)(x) (A) (f g)(x) (B) (g f)(x) (A) (f g)(x) = f(g(x)) = f(x x + 5) = (x x + 5) = x x + (B) (g f)(x) = g(f(x)) = g(x ) = (x ) FOIL (x ) + 5 = x 4x + 4 x = x 7x + 15

207 07 EXAMPLE 4 YOUR TURN 4 Let f(x) = x and g(x) = x x + 6. Find Let f(x) = x and g(x) = 5x 7x Find (A) (f g)(1) (B) (g f)( 1) (A) (f g)(1) (C) (f g)(x) (D) (g f)(x) (A) (f g)(1) = f(g(1)) = f(5) g(1) = + 6 = 5 (B) (g f)( ) = 15 f(5) = (5) = 15 (B) (g f)( 1) = g(f( 1)) = g( ) f( 1) = ( 1) = (C) (f g)(x) = g( ) = = (C) (f g)(x) = f(g(x)) = f(x x 6) = (x x 6) = 6x 9x 18 (D) (g f)(x) (D) (g f)(x) = g(f(x)) = g(x) = (x) (x) 6 = 18x 9x 6

208 08 YOUR TURN Let f(x) = x, h(x) = x 1 and g(x) = 7x 5x +. Find (A) (f f f)() YOUR TURN Let f(x) = x + 1 and g(x) = (x 1). Find (A) (f g)(1) (B) (g f)(x) (B) (g f)(0) (C) (h h h)(x) (C) (f g)(x) (D) (h h h) (x) n times (D) (g f)(x)

209 09 EXAMPLE 5 YOUR TURN 5 Find the domain of f g if f(x) = x and g(x) = x 4 Find the domain of f g if f(x) = 1 x and g(x) = x x 8 First, (f g)(x) = f(g(x)) = x 4 We know that a fraction is undefined at values which make the denominator zero; x 4 0 (x + )(x ) 0 x, Therefore, the domain is {x x, x } or (, ) (,) (, ) EXAMPLE 6 YOUR TURN 6 Find the domain of f g if f(x) = 1 4 and g(x) =. x+ x 1 Find the domain of f g if f(x) = 1 x 4 and g(x) = 1 x. First, (f g)(x) = f(g(x)) = 1 ( 4 x 1 )+ We know that a fraction is undefined at values which make the denominator zero; We consider two things; 1) ) 4 x 1 x 1 0 x 1 1 ( 4 x 1 )+ ( 4 x 1 ) x 1 4 x + 1 x Therefore, the domain is {x x 1, x 1}

210 10 EXAMPLE 7 YOUR TURN 7 Find the functions f and g such that f g = H if (A) H(x) = (x + 1) 50. (B) H(x) = 1 4x 5 Find the functions f and g such that f g = H if (A) H(x) = (x ) 4. (B) H(x) = x 1 (A) The outer function of (x + 1) 50 is ( ) 50 The inner function of (x + 1) 50 is x + 1 f(x) = x 50, g(x) = x + 1 (B) The outer function of The inner function of 1 is 1 4x 5 ( ) 1 4x 5 f(x) = 1, g(x) = 4x 5 x is 4x 5 ONE TO ONE FUNCTIONS One to one function (Invertible function): A function for which every element of the range of the function corresponds to exactly one element of the domain. HORIZONTAL LINE TEST (One to one function or not): A test uses to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to-one. fail horizontal line test pass horizontal line test YOUR TURN 8 Decide whether the following graph of a function is one to one (A) y = x + 1 (B) y = x + (C) y = x 4x + (D) y = x 1 x+1 (E) y = x 4x (F) y = x

211 11 PRACTICE PROBLEMS 1. Use the table to answer the questions x f(x) g(x) (A) (B) 1 (C) 1 (D) 1 (A) (f g)(1) (B) (g f)() (C) (f f)( ) (D) (g g)(1). Using f(x) = x x + 7 and g(x) = x, find: (A) (f g)( 1) (B) (g f)(1) (C) (f g)(x) (D) (g f)(x). Using f(x) = x x + 4 and g(x) = x, find: (A) (f g)(4) (B) (g f)() (C) (f g)(x) (D) (g g)(x) 4. Using f(x) = 1 x+ and g(x) = 5 x, find: (A) (f g)( 5) (B) (g f)() (C) (g f)(x) (D) (f g)(x) 5. Using f(x) = and g(x) = x +, find: x 1 (A) (f g)(1) (B) (g f)() (C) (g g)( ) (D) (f g)(x) 6. Find the functions f and g such that f g = H if (A) H(x) = (x ) 5 (B) H(x) = 4x 7 (A) 4 (B) 18 (C) 18x 9x + 7 (D) 6x 9x + 1 (A) (B) (C) x 7x + 14 (D) x 4 (A) 1 (B) 5 (C) 5(x + ) (D) (A) 1 (B) (C) (D) x x+5 x+ (A) g(x) = x, f(x) = x 5 (B) g(x) = 4x 7, f(x) = x (C) H(x) = 4x x+1 7. Find the domain of f g if (D) H(x) = x +5 x 6 (A) f(x) = x +, g(x) = x (B) f(x) = 1 x 1, g(x) = 1 x (C) f(x) = 1 x, g(x) = x (D) f(x) =, g(x) = 6 x 1 x x 1 (E) f(x) = x + 1, g(x) = 1 x+ (G) f(x) = x x, g(x) = x x (F) f(x) = x x, g(x) = x 1 x 4 (H) f(x) = x+5, g(x) = 1 x x 4 (C) g(x) = x, f(x) = x x+1 (D) g(x) = x, f(x) = x+5 x 6 (A) [, ) (B) (, 0) (0,1) (1, ) (C) [,) (, ) (D) (, ) (, 1) ( 1,1) (1,) (, ) (E) (, ) (, ) (F) (, ) (,4) (4, ) (G) (, ) (,0) (0, ) (H) (, ) (,1]

212 1 8. Find the following by using the graphs of functions y = f(x) and y = g(x) in the figure (A) (f + g)(1) (B) (f g)( ) (C) (f g)( ) (D) (f g)(1) (E) (g f)( 1) (F) (g f)(4) 4 y = f(x) y = g(x) 4 (A) 0 (B) 16 (C) 4 (D) (E) 1 (F) 4 9. Find the following by using the graphs of functions y = f(x) and y = g(x) in the figure (A) (f g)() (B) ( f ) ( 1) g (C) (f g)( 4) (D) (f g)(1) 4 y = f(x) (A) 4 (B) 1 (C) (D) 1 (E) (F) 0 (E) (g f)( ) (F) (g f)(4) 4 y = g(x) 10. Suppose a subway shop sells $6 per a sandwich and that x is the number of sandwiches sold per hour which is given by x(t) = t + 0t + 10, where t is the number of hours since 10 AM. (A) Find an expression for the revenue per hour, R, as a function of x. (A) R(x) = 6x (B) R(t) = 6t + 10t + 10; Revenue per hour (C) $198 (B) Find and simplify (R x)(t).what does this represent? (C) What is the revenue per hour at noon?

213 1 LECTURE 5- INVERSE FUNCTION OF A ONE TO ONE FUNCTION Video 1) Introduction to inverse function ) How to find its inverse function DEFINITION: Let f and g be functions such that f(g(x)) = x for every x in the domain of g, g(f(x)) = x for every x in the domain of f. 1) The function g is the inverse of the function f, and is denoted by f 1 (Warning: f 1 1 ) f ) The domain of f is equal to the range of f 1, and vice versa. ) If f is a one to one function, f has its inverse function and f is called an invertible function. EXAMPLE 1 YOUR TURN 1 Show that each function is the inverse of the other f(x) = x 1 and g(x) = x+1 Show that each function is the inverse of the other f(x) = x + and g(x) = x f(g(x)) = f ( x+1 ) = (x+1 ) 1 = x = x g(f(x)) = g(x 1) = x 1+1 = x = x Therefore, each function is the inverse of the other. EXAMPLE YOUR TURN Find the inverse function of the given function Find the inverse function of the given function f = {(,), (1, 4), (5,6)} f = {(,1), (1, 5), (, 7), (4,9)} The inverse is found by interchanging the coordinates in each ordered pair; f 1 = {(,), ( 4,1), (6,5)}

214 14 EXAMPLE YOUR TURN Find the inverse function of the given function Find the inverse function of the given function f(x) = x f(x) = x + 8 The inverse function of y = x is x = y x + = y x + Then f 1 (x) = x + = y Check: f 1 (f(x)) = f 1 (x ) = (x ) + f(f 1 (x)) = f( x + = x = x ) = ( x + ) = x + = x EXAMPLE 4 YOUR TURN 4 Find the inverse function of the given function f(x) = x 4 + Find the inverse function of the given function 5 f(x) = x 7 The inverse function of y = x 4 + is x = y 4 + x = y 4 (x ) = y 4 (x ) + 4 = y Then f 1 (x) = (x ) + 4 Check: f 1 (f(x)) = (( x 4 + ) ) + 4 = x = x f(f 1 (x)) = ((x ) + 4) 4 + = x + = x

215 15 EXAMPLE 5 YOUR TURN 5 Find the inverse function of the given function Find the inverse function of the given function f(x) = (x + ) 6 f(x) = (x 5) + The inverse function of y = (x + ) 6 is x = (y + ) 6 x + 6 = (y + ) x + 6 = y + x + 6 = y Then f 1 (x) = x + 6 Check: f 1 (f(x)) = f 1 ((x + ) 6) = (x + ) = x + = x f(f 1 (x)) = f ( x + 6 ) = ( x ) 6 = x = x EXAMPLE 6 YOUR TURN 6 Find the inverse function of the given function f(x) = x 4 + Find the inverse function of the given function 5 f(x) = x + 5 The inverse function of y = x 4 + is x = y 4 + x = y 4 (x ) = y 4 (x ) + 4 = y Then f 1 (x) = (x ) + 4 Check: f 1 (f(x)) = (( x 4 + ) ) + 4 = x = x f(f 1 (x)) = ((x ) + 4) 4 + = x + = x

216 16 EXAMPLE 7 YOUR TURN 7 Find the inverse function of the given function Find the inverse function of the given function f(x) = (x + 4) 5 f(x) = (x 7 + 4) The inverse of y = (x + 4) 5, is x = (y + 4) 5 5 x 5 x = y = y 5 x 4 = y Therefore, f 1 (x) = x 4 5 Check: f 1 (f(x)) = f 1 ((x + 4) 5 ) = (x + 4) 5 4 = x = x = x 5 5 f(f (x)) = f ( x 4) = (( x 4) + 4) 5 = ( x 4 + 4) 5 5 = ( x) 5 = x

217 17 EXAMPLE 8 YOUR TURN 8 Find the inverse function of the given function f(x) = 5x 4 8x + 7 Find the inverse function of the given function f(x) = x + 4x + 7 The inverse of y = 5x 4 8x+7 is x = 5y 4 8y + 7 (8y + 7)x = 5y 4 8xy + 7x = 5y 4 8xy 5y = 7x 4 y(8x 5) = 7x 4 Therefore, y = 7x 4 8x 5 f 1 (x) = 7x 4 8x 5 = 7x + 4 8x 5 Check: f 1 (f(x)) = f 1 ( 5x 4 8x + 7 ) = 7 ( 5x 4 8x + 7 ) 4 8 ( 5x 4 8x + 7 ) x 8 ( 5x ) = 8x x 40x 5 ( ) 8x + 7 = 67x 67 = x f(f 1 (x)) = f ( 7x 4 8x 5 ) = 5 ( 7x 4 8x 5 ) 4 8 ( 7x 4 8x 5 ) x + 0 ( 5x ) = 8x 5 56x + 56x 5 ( ) 8x 5 = 67x 67 = x

218 18 YOUR TURN Find the inverse function of the given function f(x) = x x 5 YOUR TURN Find the inverse function of the given function f(x) = x, x 0

219 19 NOTATION: If f is an invertible function, f(a) = b a = f 1 (b) f 1 is the mirror image of f about y = x (a point (,5) in y = f(x) has a corresponding point (5,) in y = f 1 (x) ) 4 ( 1,1) 4 (0,) y = f(x) (1,) y = f 1 (x) (,1) (,0) 4 (1, 1) EXAMPLE 9 YOUR TURN 9 Given a graph of y = f(x) in the figure, draw the graph Given a graph of y = f(x) in the figure, draw the graph of y = f 1 (x). of y = f 1 (x). y = f(x) y = f(x) Since points (, 1), ( 1,1), and (1,) are on the graph of f(x), ( 1,), (1, 1), (,1) are on the graph of f 1 (x). 4 5 y = f -1 (x) 4

220 0 PRACTICE PROBLEMS 1. Determine whether the following is a one to one function or not. (A) y = x + (B) y = x 4x (C) y = x (D) y = x (E) y = 1 x +1 (G) y = x (F) y = 1 x+1 (H) y = x x (A) Not one to one (B) Not one to one (C) Not one to one (D) One to one (E) Not one to one (F) One to one (G) One to one (H) Not one to one. Determine whether the following graph is invertible or not. (A) y (B) y (A) Invertible (B) Invertible (C) Not (D) Not x x (C) y (D) y x x. Using the graph of y = f(x), draw the graph of y = f 1 (x). (A) y = x (B) y = x y = f(x) A B y = f(x) Find the inverse function of the following function (A) f = {(,), ( 1,5), (,7), (5, )} (B) f = {(,5), ( 1,7), (0,0), (8,9)} (C) f(x) = 4x (D) f(x) = x + 6 (E) f(x) = x 5 5 (G) f(x) = x 7 (F) f(x) = x (H) f(x) = x + 8 (A) (B) {(,)(5, 1), (7,), (,5)} {(5, )(7, 1), (0,0), (9,8)} (C) f 1 (x) = x+ 4 (D) f 1 (x) = 1 x (E) f 1 (x) = x + 5 (F) f 1 (x) = (x + 5) (G) f 1 (x) = x (H) f 1 (x) = (x 8) 7 +

221 1 (I) f(x) = x + 6 (J) f(x) = (x + ) 6 (K) f(x) = (x 5) 5 (L) f(x) = (x + 7) (M) f(x) = x 1 x+ (O) f(x) = x 1 x+4 (N) f(x) = x+ x+ (P) f(x) = 7 5x 4x (Q) f(x) = x, x 0 (R) f(x) = (x ) +, x (S) f(x) = x + 4 (T) f(x) = x + 1 (I) f 1 (x) = x 6 (J) f 1 (x) = x + 6 (K) f 1 5 (x) = x + 5 (L) f 1 7 (x) = x 4 7 (M) f 1 (x) = x 1 x (N) f 1 (x) = x x 1 (O) f 1 (x) = 4x 1 x (P) f 1 (x) = x+7 4x+5 (Q) f 1 (x) = x + (R) f 1 (x) = x + (S) f 1 (x) = x 4, x 0 (T) f 1 (x) = (x 1) +, x 1

222 LECTURE 5- EXPONENTIAL FUNCTIONS AND LOGARITHMIC FUNCTIONS Video 1) An Introduction to Exponential Functions ) An introduction to logarithmic Functions ) Changing the base of logarithms 4) Converting between exponentials and logarithms EXPONENTIAL FUNCTION: An exponential function f with base a is a function of the form f(x) = a x exponent, or y = a x base, where a is real numbers such that a > 0 and a 1 is a real number. 1) Domain: (, ) ) Range: (0, ) since a x > 0 ) The graph of f will never touch the x-axis because a x > 0 4) Horizontal Asymptote: y = 0 5) y-intercept is 1. 6) f is a one to one, continuous and smooth function (no sharp points) 7) a > 1 0 < a < y = a x, a > 1 y = a x, 0 < a < 1 It is increasing As x +, f(x) + and As x, f(x) +0 It is decreasing As x +, f(x) +0 and As x, f(x) + 8) Since it is a one to one function, a t = a s t = s 9) The base 10, is often called the common base. The base e(.178) called the natural base. For now, it is enough to know that since e > 1, f(x) = e x is an increasing exponential function.

223 PROPERTIES OF LOG: Consider f(x) = log a (x) where a > 0 and a 1 f(x) = log a (x) where a is called the base, and x is called the argument. 1) Domain of f(x) = log a (x) is the interval (0, ). ) Range of f is the interval (, ). ) Vertical Asymptote: x = 0. 4) x-intercept is 1 5) f(x) = log a (x) is one to one (its inverse is f 1 (x) = a x ). 6) a > 1 0 < a < 1 y = log a(x), a > 1 y = log a(x), 0 < a < It is increasing It is increasing As x +0, f(x) and As x +0, f(x) + and As x +, f(x) + As x +, f(x) 7) Since f(x) = log a (x) is one to one (its inverse is f 1 (x) = a x ): log a x = log a w implies x = w. 8) log a (g(x)) is defined if g(x) > 0 LOGARITHMIC FUNCTIONS: For a > 0 and a 1, the logarithmic function with the base a is denoted f(x) = log a (x), where y = log a (x) if and only if a y = x (it is the inverse function of y = a x ) When a = 10, we write log 10 (x) = log(x) ; it is called a common logarithm. When a = e, we write log e (x) = ln(x) ; it is called a natural logarithm. CHANGE OF BASE PROPERTY: If M and a are positive real numbers with a 1, log a (M) = ln(m) = log b (M) ln(a) log b (a) for b > 0 but b 1.

224 4 EXAMPLE 1 YOUR TURN 1 Find the domain, range, and Horizontal asymptote of a Find the domain, range, and Horizontal asymptote of a following function. following function. y = x y = x y = x Domain: (, ) Range: (, ) Horizontal Asymptote: y = EXAMPLE YOUR TURN Find the domain, range, and Vertical asymptote of a Find the domain, range, and Vertical asymptote of a following function. following function. y = log 5 (x + ) y = log (4 x) y = log 5 (x + ) Domain: (, ) Range: (, ) Vertical Asymptote: y =

225 5 EXAMPLE YOUR TURN Write each equation in its equivalent exponent form. Write each equation in its equivalent exponent form. (A) log(x) = a (B) log (5) = x (A) log a (b) = x (A) log(x) = a log 10 (x) = a x = 10 a (B) log (5) = x 5 = x (B) log (x + ) = 5 (C) log(b) = a (D) log(x) = (E) ln(b) = y EXAMPLE 4 YOUR TURN 4 Write each equation in its equivalent logarithmic form. Write each equation in its equivalent logarithmic form. (A) x = a (B) e x = y (A) 4 = x (A) x = a x = log (a) (B) x = 5 (B) e x = y x = log e (y) x = ln(y) (C) (0.7) t = 5 (D) 10 z = y (E) e z = y 6

226 6 EXAMPLE 5 YOUR TURN 5 Find the inverse function of a following function Find the inverse function of a following function f(x) = e x + 1 y = e x The inverse of y = e x + 1 is x = e y + 1 x 1 = e y y = log e (x 1) : change it as log form y = ln(x 1) y = ln(x 1) + Then f 1 (x) = ln(x 1) + EXAMPLE 6 YOUR TURN 6 Find the inverse function of a following function Find the inverse function of a following function f(x) = log (x + 5) f(x) = log (x 1) The inverse of y = log (x + 5) is x = log (y + 5) 1 x = log (y + 5) y + 5 = x/ : change it as exponential form y = x/ 5 Then f 1 (x) = x/ 5

227 7 EXAMPLE 7 YOUR TURN 7 Describe in a sentence the relationship between the graphs Describe in a sentence the relationship between the graphs of y = 10 x and y = log(x). of y = x and y = log (x) (inverse relation) It is the mirror image of each other. EXAMPLE 8 YOUR TURN 8 Evaluate log 7 (10). Round the thousandth log 7 (10) = ln(10) ln(7) 1.18 Using the calculator, evaluate. Round the thousandth. (A) log () (B) log 6 (4) (C) log 1/ (4)