Pre-Calculus Summer Packet

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1 Pre-Calculus Summer Packet 1. Complete the attached summer packet, which is due on Friday, September 6, The material will be reviewed in class on Friday, September 6 and Monday, September 9, and Tuesday, September 10, An assessment exam will take place on Wednesday, September 11, 2013.

2 Pre-Calculus Summer Packet FINDING THE SLOPE OF A LINE FORMULA TO FIND THE SLOPE OF A LINE - THE SLOPE OF A LINE THROUGH POINTS A (x 1, y 1 ) and B (x 2, y 2 ) IS GIVEN BY THE FORMULA: m = y 2 y 1 x 2 x 1 THE SLOPE OF ANY HORIZONTAL LINE: m = 0 THE SLOPE OF ANY VERTICAL LINE: m is UNDEFINED (NO SLOPE) SLOPES OF PARALLEL LINES - PARALLEL LINES HAVE EQUAL SLOPES. SLOPES OF PERPENDICULAR LINES - PERPENDICULAR LINES HAVE SLOPES WHICH ARE OPPOSITE RECIPROCALS OF EACH OTHER. THE PRODUCT OF THEIR SLOPES EQUALS Find the slope of the line through the points (2, -1) and (4, ¾). 2. Show that the line through (2, 3) and (7, 2) is perpendicular to the line through ( 3, 7) and (2, 2). FINDING THE EQUATION OF A LINE FOUR FORMULAS FOR THE EQUATION OF A LINE STARDARD FORMULA (or GENERAL FORMULA) - Ax + By = C where A, B, and C are integers. SLOPE-INTERCEPT FORMULA - y = mx + b where m is the slope and b is the y-intercept. THE POINT-SLOPE FORMULA - y y 1 = m(x x 1 ) where m is the slope and (x 1, y 1 ) is a point on the line. THE INTERCEPT FORMULA - x_ + y_ = 1 where a is the x-intercept and b is the y intercept. a b 3. Write and equation for the line through the following points. a) a horizontal line through the point (4, 3) b) a vertical line through the point ( 2, 3) 4. Find the equation (in Standard Form) of a line that passes through the points (3, 5) and (-1, 3). 5. Find the slope and y-intercept of the line whose equation is 3y = 11x. 6. Find the slope and y-intercept of the line whose equation is y = Write the equation of a line in Standard Form with x-intercept 8 and y-intercept 4. OVER

3 Page 2 DISTANCE FORMULA AND MIDPOINT FORMULA THE FORMULA TO FIND THE DISTANCE BETWEEN TWO POINTS (x 1, y 1 ) and (x 2, y 2 ) ON A SLANTED LINE: d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 THE FORMULA TO FIND THE MIDPOINT BETWEEN TWO POINTS (x 1, y 1 ) and (x 2, y 2 ) ON ANY LINE: 8. Find the distance between the points (1, ¼) and (-1, 2). 9. Find the midpoint between the points (1, ¼) and (-1, 2). M = ( x 1 + x 2, y 1 + y 2 ) 2 2 SOLVING A LINEAR SYSTEM ALGEBRAICALLY SUBSTITUTION METHOD 1. GET ONE VARIABLE (either x or y) BY ITSELF IN ONE OF THE EQUATIONS. 2. SUBSTITUTE THE ALGEBRAIC EXPRESSION IN STEP 1 INTO THE OTHER EQUATION AND SOLVE IT. 3. SUBSTITUTE THE ANSWER IN STEP 2 INTO EITHER EQUATION AND SOLVE IT. 1. WRITE BOTH EQUATIONS IN STANDARD FORM. LINEAR COMBINATION METHOD 2. IF NECESSARY MULTIPLY EITHER OR BOTH EQUATIONS BY A NUMBER SO THAT THE SUM OF THE x TERMS OR y TERMS EQUALS ZERO. 3. ADD THE TWO EQUATIONS. (THE SUM OF THE x TERMS OR y TERMS MUST = 0.) 4. SOLVE THE ANSWER EQUATION IN STEP SUBSTITUTE THE ANSWER IN STEP 4 IN EITHER EQUATION AND SOLVE IT. 10. Solve the linear system Algebraically. 3x + 2y = -1 5x = y 11. Solve the linear system algebraically. x 3y = 4 y = 5x 8 SIMPLIFYING RADICALS SIMPLIFYING THE SQUARE ROOTS OF IRRATIONAL NUMBERS FOR ANY POSITIVE REAL NUMBER a and b: ab = a b 12, Simplify the following radicals. a) 63 b) 98 c) 75 d) 6 108

4 COMPLEX NUMBERS Page 3 COMPLEX NUMBER - A COMPLEX NUMBER IS A NUMBER WITH THE FORMULA a + bi where a and b are REAL NUMBERS and i = 1. TYPES OF COMPLEX NUMBERS 1. REAL NUMBER - A REAL NUMBER, a, IS A COMPLEX NUMBER WHERE b = 0 IN THE FORMULA a + bi. 2. PURE IMAGINARY NUMBER - A PURE IMAGINARY NUMBER, bi, IS A COMPLEX NUMBER WHERE a = 0 IN THE FORMULA a + bi. POWERS OF i i = ALWAYS KEEP i AS i. i² = 1 ALWAYS CHANGE i² TO 1. i³ = i ALWAYS WRITE i 3 AS i. i 4 = 1 ALWAYS CHANGE i 4 TO IMAGINARY NUMBER - AN IMAGINARY NUMBER, a + bi, IS A COMPLEX NUMBER WHERE BOTH a AND b ARE NOT EQUAL TO 0 IN THE FORMULA a + bi. COMPLEX BINOMIAL CONJUGATES TWO COMPLEX NUMBERS OF THE FORM a +bi and a bi ARE COMPLEX BINOMIAL CONJUGATES. PRODUCT OF COMPLEX BINOMIAL CONJUGATES THE PRODUCT OF (a + bi)(a bi) = a 2 + b 2 which is a REAL NUMBER. STEPS TO RATIONALIZE A PURE IMAGINARY DENOMINATOR 1. MULTIPLY THE NUMERATOR AND DENOMINATOR BY i. 2. SIMPLIFY THE NUMERATOR AND DENOMINATOR. STEPS TO RATIONALIZE A BINOMIAL COMPLEX NUMBER DENOMINATOR 1. MULTIPLY THE NUMERATOR & DENOMINATOR BY THE CONJUGATE OF THE DENOMINATOR. 2. SIMPLIFY THE FRACTION. 3. WRITE THE ANSWER IN a + bi form. 13. Simplify. (6 i)(6 + i) 14. Simplify. (5 + i 5 ) (5 i Simplify. (8 + 3i)(2 5i) 16. Simplify (4 5i) Simplify the expression Rationalize the denominator. 5_ i OVER

5 Page Rationalize the denominator. Write the expression in a + bi form i 20. Rationalize the denominator. Write the expression in a + bi form. 5 + i 5 i 21. Simplify the expression. i + i 2 + i 3 + i 4 + i Simplify the expressions. i Simplify the expression i 36 QUADRATRIC EQUATIONS IN ONE VARIABLE STANDARD FORMULA FOR AN INCOMPLETE QUADRATIC EQUATION ax 2 + c = 0 where a and c are real numbers STEPS TO SOLVE AN INCOMPLETE QUADRATIC EQUATIONS USING SQUARE ROOT METHOD 1. GET THE 2 nd DEGREE VARIABLE BY ITSELF ON THE LEFT SIDE OF THE EQUATION. 2. GET THE CONSTANT TERM(S) ON THE RIGHT SIDE OF THE EQUATION. 3. GET THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION. 4. IF THE SQUARE ROOT OF THE CONSTANT TERM IS A REAL NUMBER, USE BOTH THE POSITIVE SQUARE ROOT AND THE NEGATIVE SQUARE ROOT FOR THE SOLUTIONS. Solve the following equations by the square root method. Simplify all radicals. Rationalize all denominators. Express imaginary solutions in a + bi form x 2 = x 2 = x 2 2 = x = 0 STANDARD FORMULA FOR A COMPLETE QUADRATIC EQUATION ax 2 + bx + c = 0 where a, b, and c are real numbers STEPS TO SOLVE OF A COMPLETE QUADRATIC EQUATION BY FACTROING 1. MAKE THE EQUATION EQUAL TO ZERO (STANDARD FORM). 2. FACTOR THE LEFT SIDE OF THE EQUATION. 3. USE THE ZERO PRODUCT PROPERTY TO MAKE THE FACTORS EQUAL TO ZERO AND SOLVE FOR x. Solve by factoring x 2 4x 7 = x 2 8x 32 = 0

6 30. x 2 10x 1575 = 0 Page x 2 16x 1768 = x 2 8x 20 = 0 STEPS TO SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE 1. IF THE COEFFICIENT OF x 2 IS NOT 1, DIVIDE ALL TERMS IN THE EQUATION BY THE COEFFICIENT OF x TRANSPOSE THE CONSTANT TERM TO THE RIGHT SIDE OF THE EQUATION. 3. TAKE THE COEFFICIENT x AND DIVIDE IT BY 2 AND THEN SQUARE THE RESULT. 4. TAKE THE ANSWER IN STEP 3 AND ADD IT TO BOTH SIDES OF THE EQUATION. 5. FACTOR THE LEFT SIDE OF THE EQUATION AND WRITE IT AS A BINOMIAL SQUARED. 6. TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION. 7. SOLVE FOR x. THERE WILL BE TWO VALUES FOR x (POSITIVE AND NEGATIVE). Solve by completing the square. 33. x 2 8x 20 = x 2 + 6x + 10 = 0 SOLVING QUADRATIC EQUATIONS BY THE QUADRATIC FORMULA x = b b² 4ac 2a STEPS TO SOLVE A QUADRATIC EQUATION BY THE QUADRATIC FORMULA 1. MAKE THE EQUATION EQUAL TO ZERO. 2. LIST THE NUMERICAL VALUES OF a, b, and c. 3. FIND THE VALUE OF THE DISCRIMINANT b² 4ac. 4. PLUG IN THE VALUE OF THE DISCRIMINANT AND THE VALUE FOR b AND a SOLVE FOR x. Solve by the quadratic formula. Give both Real and Imaginary solutions x 2 = 7 10x 36 9x 2 12x + 4 = 121 OVER

7 Page 6 EXPONENTIAL FUNCTIONS THE LAWS OF EXPONENTS MULTIPLYING LIKE BASES: b x b y = b x + y WHEN YOU MULTIPLY LIKE BASES, WRITE THE BASE ONCE AND ADD THE EXPONENTS. DIVDING LIKE BASES: IF x IS GREATER THAN y b x = b x y b y IF THE LARGER EXPONENT IS IN THE NUMERATOR WHEN DIVIDING LIKE BASES, WRITE THE BASE ONCE IN THE NUMERATOR AND SUBTRACT THE EXPONENTS. IF y IS GREATER THAN x b x = 1 b y b y x IF THE LARGER EXPONENT IS IN THE DENOMINATOR WHEN DIVIDING LIKE BASES, WRITE THE BASE ONCE IN THE DENOMINATOR AND SUBTRACT THE EXPONENTS. NEGATIVE EXPONENT RULE: IF b 0, THEN b x = 1 or 1 = b x b x b x TO MAKE A NEGATIVE EXPONENT POSITIVE, FLIP OVER THE BASE. ZERO EXPONENT RULE: IF x = y, THEN: b x = bº = 1 (b 0) b y ANY BASE (EXCEPT ZERO) WITH A ZERO EXPONENT EQUALS 1. POWER RULES: (b x ) y = b xy WHEN YOU HAVE A MONOMIAL INSIDE A PARENTHESES AND AN EXPONENT OUTSIDE THE PARENTHESES, MULTIPLY THE OUTSIDE EXPONENT BY EACH INSIDE EXPONENTS. (ab) x = a x b x WHEN YOU HAVE A MONOMIAL INSIDE A PARENTHESES AND AN EXPONENT OUTSIDE THE PARENTHESES, MULTIPLY THE OUTSIDE EXPONENT BY EACH INSIDE EXPONENT. IF b 0 then a x = a x _ b b x WHEN THERE IS A FRACTION INSIDE A PARENTHESES AND AN EXPONENT OUTSIDE THE PARENTHESES, THE OUTSIDE EXPONENT IS MULTIPLIED BY EACH INSIDE EXPONENT.

8 RATIONAL EXPONENT PROPERTIES IF q IS A POSITIVE INTEGER AND b IS A REAL NUMBER, THEN: b 1/q = b Page 7 IF p AND q ARE POSITIVE INTEGERS, AND b p IS A REAL NUMBER, THEN: b p/q = ( b ) p or b p 37. Simplify by using powers of the same base and no negative exponents. a) b) Write each expression using a radical and no negative exponents. a) x 2/3 b) 5 ½ x ½ 39. Write each expression using positive rational exponents. a) x 5 _ b) ( 2a ) Simplify a) 9 ½ 25 b) 9 ½ 25 EXPONENTIAL FUNCTIONS y = (x) = ab x where a and b are any positive real numbers (except b 1) and x is any real number, is an EXPONENTIAL FUNCTION. a is the y- intercept, b is the base, and x is the exponent. PROPERTIES OF THE GRAPH OF (x) = ab x 1. The domain is {all real numbers} 2. The range is {y: y > 0}. 3. The graph is always increasing: if x gets larger, then y gets larger. 4. The y-intercept on the graph is (0,a). OVER

9 Page 8 y = (x) = ab x, where a and b are any positive real numbers (except b 1) and x is any real number, is an EXPONENTIAL FUNCTION. a is the y-intercept, b is the base and x is the exponent. This functions may also be written as (x) = a(1/b ) x. PROPERTIES OF THE GRAPH OF (x) = ab x 1. The domain is {all real numbers}. 2. The range is {y: y > 0}. 3. The graph is always DECREASING: if x gets LARGER, then y gets SMALLER. 4. The y-intercept on the graph is (0,a). 41. Graph (x) = 10 x in your graphing calculator and then answer the questions. a) What is the domain of (x) = 10 x? b) What is the range of (x) = 10 x? c) Is the function (x) = 10 x always increasing or always decreasing? d) What is the y-intercept of (x) = 10 x? e) What is the name given to the function (x) = 10 x 42. Graph (x) = 10 x in your graphing calculator and answer the questions. a) What is the domain of (x) = 10 x? b) What is the range of (x) = 10 x? c) Is the function (x) = 10 x always increasing or always decreasing? LOGARITHMIC FUNCTIONS f(x) = log b x where x is the POWER and any positive real number, b is the BASE and any positive real number except 1, and y is the EXPONENT and any real number. meaning: Exponent = log Base Power The COMMON LOGARITHMIC FUNCTION is written y = log x where the base 10 is understood. The NATURAL LOGARITHMIC FUNCTION is written y = ln x where the base e is understood. PROPERTIES OF LOGARITHMS Let b and x be positive real numbers such that b log b 1 = 0 because b 0 = 1 2. log b b = 1 because b 1 = b 3. log b b x = x because b x = b x

10 Page GRAPH y = logx in your calculator a) What is the domain? b) What is the range? c) Is the function always increasing or always decreasing? d) What are the coordinates of the x-intercept? 44. Write the following Logarithmic Equations in EXPONENTIAL FORM. a) log 4 16 = 2 b) log 6 (1/36) = 2 c) log 4 8 = Find each logarithm. Show your work by writing the expression in exponential form. a) log 10,000 b) log 2 4 c) log 3 27 d) log e). log 5 1

( ) c. m = 0, 1 2, 3 4

( ) c. m = 0, 1 2, 3 4 G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are: