Summer Packet A Math Refresher For Students Entering IB Mathematics SL

Transcription

1 Summer Packet A Math Refresher For Students Entering IB Mathematics SL

2 Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school year. The packet will be collected on the first day of school and given homework points. The topics in this packet are all skills that you have already seen, and it is necessary that you are comfortable working problems of these types upon entering Precalculus. Use the provided example problems and your notes from previous classes to assist in completion. Functions Operations with Functions f Example: Find ( f + g)( x), ( f g)( x), ( f g)( x), (x) for: = x + x 4 and g ( x) = x. g ( f + g)( x) = + g( x) ( f g)( x) = g( x) = ( x + x 4) + (x ) = ( x + x 4) (x ) = x + 6x 6 = x + x 4 x + = x ( f g)( x) = g( x) = ( x + x 4)(x ) = x (x ) + x(x ) 4(x ) = x x + 9x 6x 1x + 8 = x + 7x 18x + 8 f g ( x) = g( x) x + x 4 =, x x Practice: Perform the following operations for: = x + 1 and g ( x) = x. 1. ( f + g )(x). ( g f )(x). ( f g )(x) g 4. ( f g )(x) 5. (x) f f 6. (x) g Evaluating Functions Examples: m ( x) = x x + r ( x) = x + x + 1 m () = () () + r ( 4) = ( 4) + ( 4) + 1 = 8 = 61 Practice: Evaluate for: c ( x) = 16x 4x +, h( x) = x 4 x, k ( x) + x = x h(-) 8. k() 9. c 10. What does f()=5 mean? 1

3 Examples: Factor. GCF: 4x y + 6x y x = x ( x y + xy 1) Factoring Difference of Squares: 4x 9 = ( x ) ( ) x x + = ( )( ) Product/Sum: x + x 15 x + 5 x (or reverse FOIL) = ( )( ) By Grouping: x 4x + x 8 x 4x + x = ( ) ( 8) = x ( x 4) + ( x 4) = ( x 4)( x + ) Master Product: x 7x + (or just guess and check) x 6 + (-7), 5 -, , 6 7-1, x 48 = x 1x 6x + = ( x 1x) + ( 6x + ) = x ( x 1) + ( x 1) = ( x 1)( x ) x (-15) + () -1, , , - 5, - + Use ( )+( ) to group Factor out the GCF of each group Factor out the common (_) out of each term In ax + bx + c form, multiply a x c Find # s that multiply to axc and add to b Replace the middle term Factor by grouping. Practice: Factor the following completely. 1. 5a b + 10ab. x 5. x + 6x x 5. 6x 9x + x 6. 5 x 0 7. x 8x x 7x + 9. x x x x 1

4 Solving Quadratic Equations Examples: Solve the following quadratic equations. a.) Solve by factoring: x + x 15 = 0 ( x )( x + 5) = 0 x = 0 x + 5 = 0 x = x = 5 b.) Solve by using the quadratic equation: x x 6 = x x 9 = 0 a =, b =, c = 9 + ± x = ( ) 4( )( 9) ( ) ± 81 x = x = = = x = = = x =, Factor the equation Set each factor = to 0. Solve for x. Set up in ax + bx = c = 0 form. and determine a,b,c Substitute into the quadratic equation: Simplify. b ± x = b 4ac a c.) Solve by using the quadratic equation, simplify the radicals: x 4x = 8 x 4x 8 = 0 a = 1, b = 4, c = ± x = 4 ± 48 x = Practice: Solve. ( 4) 4( 1)( 8) ( 1) 4 ± 4 x = x = ± 1. x + 7x + 1 = 0. 5x = 10x. x + x = x + x + 10 = 0 5. x 16 = 0 6. x x = 0

5 Shifting Graphs Example: Be familiar with the graphs of common functions. Clearly shift points. Vertical shift c units upward: h(x)=f(x)+c Vertical shift c units downward: h(x)=f(x)-c Horizontal shift c units to the right: h(x)=f(x-c) Horizontal shift c units to the left: h(x)=f(x+c) = x = x + = x + Practice: 1. = x ( x) = x f = ( x ) + 1. = x = ( x + 4) = x. = x = x + 4 = x + 4

6 Piecewise functions: Graphing and Evaluating Piecewise functions are composed of two or more functions. Given the function: x +, x 1 = To the left of x = 1, the graph is the line y = x +. x + 4, x > 1 To the right of x = 1, the graph is the line y = x + 4 Evaluate: f ( ) = ( ) + = f ( ) = + 4 = f ( ) = ( ) + = 1 f ( ) = + 4 = 1 f ( 1) = ( 1) + = 1 f ( 4) = = 0 f ( 0) = (0) + = f ( 5) = = 1 f ( 1) = (1) + = 5 Use the first piece Use the second piece To graph, use the points from above x y x y - - y **Open Dot** 1 x ** You must evaluate the break point in each piece! One point will be a closed dot and the other will be an open dot, Evaluate and graph. 1. x +, x < 0 = x y. 5 x, x 0 x + 1, x 1 = x y x +, x > 1 5

7 Domain and Range Given a function y = f(x), the Domain of the function is the set of permissible inputs (x-values) and the Range is the set of resulting outputs (y-values). Domains can be found algebraically; ranges are often found graphically. Domains and Ranges are sets. Therefore, you must use proper set notation. When finding the domain of a function, ask yourself what values can't be used. Your domain is everything else. There are simple basic rules to consider: * The domain of all polynomial functions is the set of real numbers. * Square root functions cannot contain a negative underneath the radical. Set the expression under the radical greater than or equal to zero and solve for the variable. This will be your domain. * Rational functions can not have zeros in the denominator. Determine which values of the input cause the denominator to equal zero, and set your domain to be everything else. Examples: Find the domain of the function algebraically = Domain: x. x = = Domain: x 0, 4 x 4x x( x 4). x x = = Domain: x, x 9 ( x + )( x ) 4. = x 9 Domain: x 9 5. = 4 x Domain: x 4 6. = x 1 Set Radicand equal to zero and solve. x 1 = 0 ( x + 1)( x 1) = 0 x = 1 & x = 1 Now use a number line and decide where values will be positive Domain: x 1 & x 1 ** A number line can be used for all square root equations by testing a number in each region.** 6

8 Domain and Range Continued Identifying the domain and range from the graph is easy. For domain values, take the x-values from left to right. For range values, take the y-values from bottom to top. 1. Domain: 5 < x Domain: 4 < x 1 Domain: 6 < x < 6 Range: 4 < y 6 Range: y < 5 Range: < y < 5 Practice: Identify the domain for each function. 1. x =. x + x 1 g ( x) =. h ( x) = x 4 x + 4. = x + x 4 5. g( x) = 5 x 6. h( x) = 1+ x 7. = 8. g ( x) = x + 9x x x 6 x h ( x) = x 4 Practice: Identify the domain and range for each function y x

9 Unit Circle (, ) (, ) (, ) Key (, ) (, ) (, ) (, ) (, ) Cos θ, Sinθ (, ) o 0 (, ) Degrees (, ) (, ) (, ) (, ) (, ) (, ) (, ) Radians 8

10 Solving Absolute Value Equations 1. Solve 5x =. Check your solutions.. Solve 5x + 4 = 8 x. Check your solutions. Relations and Functions. Given the function f x = x 7x +, find: a. f( 4) b. f(n + 5)

11 Linear Equations Slope: m = y y 1 x x 1 Standard Form: Ax + By = C (remember, no fractions or decimals) Point-Slope Form: y y 1 = m(x x 1 ), where (x 1, y 1 ) are the coordinates of a point on the line and m is the slope of the line. Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept 4. Find the slope of the line containing the points (-5,4) and (6,-9). 5. Determine the slope of the line 7x 4y = 1 6. Find the standard form of a line that contains the point (5,-7) and has a slope m = Find the slope-intercept form of a line passing through the points 4,1 and (5,)

12 Parallel and Perpendicular Lines 8. Write the slope-intercept form of the equation of a line passing through the point (,1) and perpendicular to the line 4x y =. 9. Write the slope-intercept form of the equation of a line passing through the point (1, -1), parallel to the line passing through the points (4, 1) and (, -). Simplifying Radicals a b c 4

13 Special Functions Graph the functions:

14 Systems of Equations 14. Solve the system of equations: 15. Solve the system of equations: 16. Solve the system of equations: x y = 7 x + y = 8 5x + y = 8 4x + y = y = x 7 4x = y

15 Factoring 17. x x x 8x x + 5x x 4x 4 1. a 4ab + 4b. 6x 100y. 7x x 18x 7

16 Solving Equations by Factoring Solve the equations by factoring: 5. 6x x = 0 6. x + x 0 = x 5x 4 = x = 18x 6 Simplifying Rational Expressions Simplify the rational expression. State the excluded value. 9. x 4 x +6x+8 0. x +7x+1 x +x 8 8

17 Multiplying Rational Expressions Multiply the rational expressions: 1. x 64 x+16 x+8 x +16x+64. x +7x+1 x +x 1 x +x 8 x +x 8 Multiplying Rational Expressions Divide the rational expressions:. 4. 1c d c d 5a b 10ab x 9 x +1x 7 x+ 4x 1 9

18 Adding and Subtracting Rational Expressions x+ + x 1 x +x+1 x 1 4 5x 4x 4x+1 0x 5 10

19 Rational Equations Solve each equation: x+1 m+ 5m x 5 4 = 1 + m 1 m = 4 Multiplying Monomials 9. 5xy 4x ( y 4 ) 40. (x y z 5 ) ( xy z) 11

20 Dividing Monomials (xy ) z 4 x 1 y z 7 ( mn ) 4m 6 n 4 Multiplying Polynomials Multiply the polynomials: 4. x (x 4x + ) 44. x x + 1 (x x 4) 1

21 Synthetic Division Use synthetic division to divide: 45. x 7x + 9x 14 (x ) 46. x 4 4x + x + 7x (x + ) Identifying Rational Zeros List all of the possible zeros of each function: 47. f x = 6x 7 x 4 + 1x + 18x 9x f x = x 8 6x 5 x 4 + x + 4x 10 1

22 Roots and Zeros Find all the zeros of the function: 49. f x = x 10x + 4x f x = x 4 x + 1x 75x

23 Quadratic Formula: Given ax + bx + c = 0, then Honors Precalculus Summer Review Packet Reference Information b ± b 4ac x =. a Factoring: a + b = ( a+ b)( a ab+ b ) a b = ( a b)( a + ab+ b ) Laws of Exponents: ( a ) = a m n mn n a a = b b Forms of Equations of Lines: General ( Standard ) Form: n n a a n n m = a ( ab) n = a n b n m a n 1 n a m = n n m n a = a or( a) Ax + By = C Slope-Intercept Form: y = mx + b m Point Slope Form: y y = m x = ) Vertical Line: 1 ( x1 Distance Formula (given points x, ) and x, )): x = a Horizontal Line: y = b ( 1 y1 ( y d = 1 x) + ( y1 ) ( x y a + c b + d Midpoint Formula (given points ( a, b) and ( c, d) ): M, Changing between Logarithmic and Exponential Form: Basic Properties of Logarithms: log 1= 0 b log b b = 1 b b y = y y = log b ( x) iff b = x log b x y log b x = Properties of Logarithms: Product Rule: log ( RS) = log R + log b Quotient Rule: logb Power Rule: R S R = log c b b log = c log Imaginary Numbers: i = 1 i = 1 Complex Number written in Standard Form: a + bi b R log Properties of Absolute Value: a 0 a = a ab = a b Sequences and Series b R b b S S a = b a b ( Arithmetic Series: a n = a 1 + (n 1)d, S n = n a + a 1 n ) 1 r n Geometric Series: a n = a 1 r (n 1), S n = a 1 1 r n 1 = n, i=1 n i = i=1 n(n +1) n, i = i=1 n(n +1)(n +1) 6 Summer 014

24 Honors Precalculus Summer Review Packet SHOW ALL WORK ON A SEPARATE SHEET OF PAPER. Simplify each expression ( i 7) x 4x y y 5. ( + i) + (5 + 7i) 6. i ( - i) ( + i) ( - i) 8. ( 5) + i 9. 8 i ( + 9)(6 + 5) Factor each polynomial completely. 1. t 4t x x 7x 15. x x 4x 8 Simplify each expression (5x )(x ) 17. n 18. ( t )( t ) 19. ( c ) Solve each quadratic equation. 0. ( x 1)( x + ) = 0 1. x( x 4) = (4 x). x + 4x =. x x = 0 Graph the functions using a table of values, symmetry, rational zero theorem, or other properties of polynomials to plot points. Verify the graph with the calculator. Describe the following characteristics for each function: a. domain and range b. zeros c. y-intercept d. end behavior e. intervals where the function is increasing and/or decreasing 4. = x x + x g ( x) = x + x = x + 4 Summer 014

25 Honors Precalculus Summer Review Packet Given = x 4and g ( x) = x + 4, determine each of the following. 7. f () 8. = 0, when x =? 9. f (g(4)) 0. f ( g( x)) 1. Domain of f ( g( x)). f 1 ( x). Is the inverse of f (x) a function? If not, how could the domain of f (x) be restricted to make its inverse a function? Simplify each and write your answer as a single fraction. State any restrictions on the variable. x 5 x 5x + 6 x + x x + 7x+ 10 x 4 x x x 6. x + 5 x x 7. 6x x x+ x +10 Solve each equation for y y + 6x = y 7x= 4 Find the solution(s) of the given systems of equations. Write answers in the form (x, y). 40. x 5y = 7 7x + y = x + 9y = x + 6y = 1 Solve for the missing side of the triangle using the Pythagorean Theorem, a + b = c. 4. a = 6 ft., b = 8 ft. a 4. b = 17 ft., c = 19 ft. c b Solve for x and y using a (ratio of sides 1:1: ) or a triangle (ratio of sides 1: : ) x 45 x y x 4 in 60 y 0 4 cm y ft Summer 014 4

26 Honors Precalculus Summer Review Packet Given the right triangle, determine the trigonometric ratios. B sin A 48. cos A 49. tan A 50. sin B 51. cos B 5. tan B C 15 A Use trig ratios to solve for x and y (to the nearest thousandth) in each right triangle x 0 y 18 1 x 8 y Evaluate each logarithmic expression without a calculator. 55. ln 1 e log log log 5 5 Solve each equation or inequality. 59. x 1 + = x + 4 > x < 10 Find an equation in slope intercept form for the line described. 6. The line through (, - ) with slope m = 4/5 6. The line through the points ( -1, -4 ) and (, ) 64. The line through ( -, 4 ) with a slope m = The line through (, - ) and parallel to the line x + 5y = 66. The line through (, - ) and perpendicular to the line x + 5y = Summer 014 5

27 Honors Precalculus Summer Review Packet Find the distance between the two points. Then find the midpoint of the segment joining the two points. 67. (-4, -), (1, 1) Write a recursive rule for the sequence 68., 1, 7, 4, , -, 9, -7,.. Find the sum of series () i 1 i= i 9 i=1 Write the first 4 terms of the sequence 7. a n = 6 n 7. a 1 = 4 a n = 5a n 1 Write the nth term of the sequence 74. 5, 11, 17, , 7 10, 8 15, 9 0,... Solve the following quadratic system of equations 76. 4x + y = 16 x + y = 77. x + 4y 8y = 4 y y 8x 16 = 0 Graph the function and label the axes and mark the x and y intercept. State their domain and range. 78. y = x y = 1 e x 80. y = log( x + ) + Condense the following expressions 81. ln 7 + 5ln ln 4 8. log 4 + 5log 4 Are the following functions one to one? If not, can you restrict the domain so that it is 1:1. Find the inverse of the following functions. Graph the function and its inverse on the same set of axes. Label each graph. 84. f (x) = 7 x f (x) = x 5 6 Summer 014 6

28 Pre-Calculus Summer Assignment The following worksheets represent your first week of work for Pre-Calculus. It should all be review of topics you learned in Advanced Algebra except for the last page which is a little more challenging. If you do not remember how to do some of these problems, watch a video on khanacademy.org (Algebra section) or a you tube video to help you remember the curriculum. This packet is due on the first day of math class which will be the week of August 5 th. Pre-Calculus Name: Assignment #1 - Sections 1. and 1.4 Solving Equations and Lines Solve the following equations. Show your steps!! 1. 7 (4x ) ( x 4). 4x (5x ) ( x 1) 7x x 1 x. 5 7 x x 5. x x 1 x x 6. x x1 7. ( x 7)( x 1) ( x 1) 8. x 4x 4

29 9. 5x 6 1 x 10. x x 9x x x 1. x x Find an equation for the line with the given properties. Express your answer in slope/intercept form y mx b 1. Slope = ; containing the point (4, ) 14. Slope = ; containing the point (1, 1) 15. Containing the points (,4) and (,5) 16. x-intercept = ; y-intercept = Horizontal; containing the point (,) 18. Vertical; containing the point (4, 5) 19. Parallel to the line y x ; containing the point ( 1,) 0. Parallel to the line x y 4; containing the point ( 5,) 1. Perpendicular to the line y x ; containing the point (1, ). Perpendicular to the line x y 4; containing the point (,4)

30 Pre-Calculus Assignment # Section.1 Functions Find the following values for each function. 1. x x 1 a) f (0) b) f ( 1) c) f ( x) d) f (x) e) f (x 1). Name: x x a) f (0) b) f ( 1) c) f ( x) d) f (x) e) f (x 1) 1 4 f) f) Find the domain of each function. x. 4. x 4 x x 4 4x 5. x 1 Find the following values for the given functions. 6. x and g ( x) 4x 1 a) ( f g)( x) b) ( f g)( x) c) ( f g)( x) d) ( f g)() e) ( f g)( 4) 7. 1 f( x) 1 and x a) ( f g)( x) b) ( f g)( x) c) ( f g)( x) d) ( f g)() e) ( f g)( 4) gx ( ) 1 x

31 f ( x h) Find the difference quotient using the equation h 0. (example below) h Example: x x 4 f ( x h) ( x h) ( x h) 4 x xh h x h 4 Therefore, f ( x h) h x xh h x h x x 4 ( 4) h The answer is x h 1 xh h h h x h 1 8. x 1 9. x 5x x 5x 7

32 Advanced Algebra Summer Assignment The following worksheets represent your first week of work for Advanced Algebra. It should all be review of topics you learned in Algebra except for the last page which is a little more challenging. If you do not remember how to do some of these problems, watch a video on khanacademy.org (Algebra section) or find a video on you tube on that subject. This packet is due on the first day of math class which will be the week of August 5 th. Assignment #1 - Linear Equations Name: Show your work. Leave all answers as fractions when appropriate. State whether each equation is linear: 1. y 7 x. y x. x7y y x x 5 Graph each linear equation. Graph #5 and 6 on the left grid and #7 and 8 on the right grid y x 4 6. y x 7. y x 8. 4 y x 5 Determine whether each table represents a linear relationship between x and y. If so, write the next ordered pair that would appear in the table and find the equation of the line in y mx b form. Example: 1. Look for addition patterns in x and y. x changes by + and y changes + y. Find the slope m x. Find the y-intercept by finding the ordered pair with x = 0. (0,1) x y The equation is y x 1 9. x y x y x 4 5 y x 1 4 y

33 y y1 Find an equation for the line with the given properties. Use the slope equation m and x x 1 either y mx b (slope/intercept) or y y mx x (point/slope). Express your answer in slope/intercept form y mx b 1. Slope = 4; containing the point (, 1) Slope = ; containing the point (6, ) 15. Containing the points ( 4, ) and ( 5,) 16. x-intercept = ; y-intercept = Horizontal; containing the point (, ) 18. Vertical; containing the point (4, 5) 19. Parallel to the line y x 5; containing the point ( 1,7) 0. Parallel to the line xy 6; containing the point ( 6,) 1. Perpendicular to the line yx 5; containing the point (8, 7). Perpendicular to the line 4x y 9; containing the point ( 6,1)

34 Graph each of the following equations given in standard form using intercepts (see example). Example: Graph the two points and sketch a line through them Graph xy 6 using intercepts. Step 1: Find the y-intercept by making x = 0 0 y 6 y 6 y intercept (0, ) Step : Find the x-intercept by making y = 0 x 0 6 x 6 x intercept (,0). xy 4 4. x 4y x y x y 10

35 Assignment # - Solving Equations and Inequalities Name: Solve for x. Show your work and leave all answers in fractions when appropriate. Hint: To solve equations with fractions, multiply all terms by the least common denominator. 1. 4(x 4) (7x ) 0. (6x 5) 5(9 4 x). 7( x 4) x 4( x 1) 4. 5 x (x 9) 4x x 6. 1 x x 5 x 8. 1 x 9 x 4 9. x ( x1) 4 (x1) 1

36 Solve each literal equation for the indicated variable. Use the same methods that you would use for solving an equation with numbers. 11. A ( a b) h for h 1. A 1 1 h( b b ) for h 1. ax b cx d for d 14. ax b cx d for x 15. I P( 1 rt) for r 16. T T0 a( z z0) for a for r 18. R r r 1 u 1 y for u u