Name: Period: Date: COLLEGE PREP ALGEBRA 1 SPRING FINAL REVIEW

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1 Name: Period: Date: MODULE 3 COLLEGE PREP ALGEBRA 1 SPRING FINAL REVIEW Unit 5 Sequences and Functions 1. Write the first 5 terms of each sequence, then state if it is geometric or arithmetic. How do you know? O Exponential a. f( n ) = n for n 0 Linear Explicit Formula Geometric Linear Adding/subtracting b. f( n) = 6n+ 4for n 0 Explicit formula Arithmetic Linear Recursive formula c. f( n+ 1) = f( n) + 3, f(1) = 4 for n 1 Arithmetic Exponential d. a 3 n n = for n 1 Explicit Formula Geometric. Given the following sequences, find f() and state if the sequence is geometric or arithmetic and how you know. a. 8, 10, 1, 14, 16 b. 3, 6, 1, 4, 48 fe t Arithmetic because add io c. f( n ) = 4 n d. f( n) = n 3 Geometric because It s exponential multiply by 4 f ()= 16 O 3. What is the difference between a recursive and an explicit formula? Recursive tells the relationship between current term f(n) and next term f(n+1), you need first term. Explicit formula is a formula that you plug in the term value. 4. Write an explicit and a recursive formula for the following sequence: Term: ** **** ****** ******** Term #: Multiply/divide Exponential FEET Geometric because Multiply by felt f Arithmetic because Linear/ subtract by f()= -()-3=-4-3=-7 fan N Explicit: flnty fn 1 Recursive:

2 x 3 x 5. Given f( x) = 1 x+ 7 x> evaluate the function at the following x-values: a. f(0) b. f() c. f(6) E 6. List the domain and range of each relation below. Decide if each relation is a function why or why not? a. {(1, ), (3, 4), (5, 6)} b. {(1, 1), (, 1), (3, 1)} c. d. relationship between number of hours studied and grade earned on the test e. k 34,5 k f( t) = 16t + 68t+ 60 represents the height of a ball thrown vs. time in the air 7. Decide if each statement represents a function, yes or no. a. The assignment of principals to schools. b. The assignment of car companies and car models. c. The assignment of students to their math class (assuming each student takes one math class). d. The assignment of math classes to a student (assuming each student take one math class). e. The assignment of pro-football players to an NFL team. f. The assignment of NFL teams to their players. 8. Explain how you know if a graph is a function. Not Function: one x value matches with multiple y value. CANNOT have same x value for different y value.3 Not st D of hours R grade Domain Y µ14 Use vertical line testing. If two or more points on the vertical line, then it s NOT function.

3 Use the graph to the right to answer # f A b 9. Domain 10. Range k(-) D V f y 1. k(3) 13. k(0) k(-8) Z 15. k(-3) Z 16. Interval(s) where increasing E O D 17. Interval(s) where decreasing 18. Interval(s) where constant Unit 6 - Exponential Functions 19. On the accompanying grid, sketch the graphs of and. Include several key points on each graph and any other key features. Identify the name of the type (family) of each function and the coordinates of all point(s) of intersection. i 3 o t 3 Type of function: Horizontal shift when added to x Type of function: x f(x) x g(x) Lo 1 th 4 Point(s) of Intersection: I Label the following as linear decay, linear growth, exponential decay, or exponential growth. a. Amount of medication remaining in the body over time. b. Population of fish in a pond. c. March Madness playoffs that end in 1 winner. d. Adding $50 to your bank account each month. (cont. next page)

4 4 e. f. g. 3 h. i. 1. If you are starting a rare stamp collection and you buy 3 new collectable stamps each month, is this situation linear or exponential growth? Write the equation you would use to find how many stamps you would have after m months. 3 vn fcmf. If you tell a rumor to people on the first day, and those people each tell more people on the second day, who then tell more people on the 3 rd day, does this represent linear or exponential growth? When will more than 50 people know the rumor not including yourself? fit zt f41 fly 4 ft 3 8 fit 4 16 fsu 5 3 fd z Lori s car value decreases by 5% each year. If she bought the care for $3000, after how many years will it be worth less than $1000? Flt 3ooo to 4. Identify whether each table contains pairs of values that could be modeled by an exponential function, linear function, quadratic function, or none. a. X Y b. X Y c. X Y d. X Y vertex K fts f It J z linear linear quadratic exponential

5 Use the exponential growth and decay formulas to answer the following questions. Exponential growth: Exponential decay: 5. In 000 the population of deer in a local forest was approximately 1,100. If the population decreases at a rate of 4%, write an expression and find the deer population five years later. I 100 I O 04ft fl 51 1too Joe borrows $500 at 8% interest. Write an equation to represent the amount of money f(t) that Joe will owe after t years. f E 500 I too l o 8 t 7. Mary invests $000 at.6% interest compounded annually. Write an equation to represent the amount of money f(t) that Mary will have in the account after t years. ft 000 Ito o 6 t 8. A certain radioactive element decays over time according to the equation, where A = the number of grams present initially and t = time in years. If 9000 grams were present initially, how many grams will remain after 400 years? 9. Which equation models the data in the accompanying table? J 9 ooo f Time in hours, x Population, y a. y = x + 5 b. y = x c. y=x d. y = 5() x linear if 0 linear go oo gt 1000 grams 30. Judy works for a doctor. She placed a sample of bacteria in a culture dish and recorded the number of bacteria present each 30 minutes beginning at 1:00 P.M. The table shows Judy s data. If the pattern of bacterial growth remains constant, how many bacteria should be present in the culture dish at :00 P.M.? Bacterial Growth Number of Time Bacteria Present 1:00 P.M :30 P.M :00 P.M. 400 t 3 o 3 41 ti

6 Graph each exponential funtion. 31. x y = 3. 1 x y = x y = x y = 35. If the equation 5 x y = is graphed, which of the following values of x would produce a point closest to the x- axis? a. 3 b. 0 c. 3 d. 5 t f f 5 x

7 number (a)in front of the function is shrink or stretch a> 1 stretch. a<1 shrink. - flip over x-axis number added to x/ opposite Horizontal shift: + left; - right Unit 7 Transformations of Functions and Modeling 36. Describe the transformations of the absolute value graph described by the equation kx ( ) = x number added to y/ same Right 3 down 4 Vertical shift: + up; - down 37. Write the equation of the absolute value function that stretches by a factor of, shifts horizontally to the left 5 and vertically up 9. f ,3 # The graph of f(x) is given to the right. Use it to match each of the transformations to the appropriate equation below. Describe the transformations shown in each function. One Ii'd equation will not be used. Justify each answer. gx ( ) = f( x 1) + 3 hx ( ) = f( x+ 1) + 3 jx ( ) = f( x+ 1) 3 kx ( ) = f( x 1) + 3 Horizontal shift= opposite yl p ups right 41. If some values of f(x) are shown in the table below, write a new table with the following transformations: a) g(x) = -f(x) 3 b) h(x) = f(x-1) + i

8 MODULE 4 Units 8-10: Quadratics In #4-50, Factor Completely. 4. x 5 IIIzaltbka b 43. x 10x+ 4 Kt5X x 19x x8X 45. 3x 18x xy x y + xy Fix.FI 3 y E ieizet 48. yy yy 49. k+ 3k L4i4Yt49 I 51. Given the quadratic equation, ff(xx) = xx 4xx 1, identify the zeros. 47. xy + 6xy + 8xy x 17x + 7x 5. Given the quadratic equations, write them in vertex form by completing the square, then verify that you did it correctly by putting them back into standard form. a. y x x xh3 = b xtai Ex 6 GAY 1 6 x f b b b ElbZ4b1t3y ( ) = XXZl7Xt7 xcxsx exormeiej4 b i 3 check L xf fcbj zlb5 bz8bt8 ge lbz4bt4 5 exflb

9 I l 53. Given the quadratic equation kk(xx) = (xx + 4) 9, identify the translations when compared to the parent graph yy = xx. Vertical stretch, left 4, down Find the roots of the following quadratic equation, (xx + ) = 36. te F O 55. Given the quadratic equation, yy = (xx 7) + 8, identify the Axis of Symmetry and Vertex. 56. Use any method to find the vertex and solutions/roots/zeros. yy = xx 6xx Solve 4( x + ) = 0. Vertex 58. Find the vertex of the following equations, then state how many solutions/roots/x-intercepts they would each have how do you know? a) f(x) = (x-3)(x+) b) 59. Sketch a graph of a quadratic that has: 7,8 A O S X y y x46xt y ye ex x Nxtzz LXty tsfxtz.t.rs x z Vertext35 gx x x ( ) = x 3x z xz3x194 8 I two solutions X f a) x-intercepts b) 1 x-intercept c) 0 x-intercepts bz4ac X I x 3, HE X tz E.E Vertex I Nosolutions Vertex l's,54

10 a b 4c 60. How many x-intercepts does gx ( ) = x 4x+ 1have? How do you know? 5 49C Two X intercepts 61. Given the graph, write the function in vertex form of the following translation compared to the parent function yy = xx. y kxt3 4 Left 3, down 4 t Graph the quadratic equationyy = xx 4xx 1? 11 z xzzx 11 l l l 63. Write a quadratic equation that would show a transformation from the parent graph by being vertically shrunk by 1, shifted 7 to the right, and down zixifd I 1 15 I y ztxt Z Write a Quadratic Equation that would show a transformation from the parent graph by being vertically stretched by 4, shifted to the left, and up

11 65. Sketch a parabola that shows a vertex of (4,5) 66. Sketch the parabola y = x + 6x+ 8 fix Describe the transformation from the parent graph of the following Quadratic Equation: Vertical stretch by, right 4; up 5 y = + ( x 4) Given the graph of the quadratic function below, answer the questions: Time (seconds) a. What is the domain of the function graphed? b. What is the range of the function graphed? o c. At what time does the 45 ball hit its maximum? 0 Sec d. What is the maximum height that the ball reaches? 16 e. If the graph continued feet to the left, where would the other zero be? 5 f. Where is the y-intercept? g. At what interval is the graph decreasing? 0 45 h. When is the height of the ball about 11 feet? 6 34

12 69. Zach throws a hockey puck in the air with an initial velocity of 48 ft/sec from an initial height of 6 feet. The quadratic model that represents this situation is ff(tt) = 16tt + 48tt + 6. a. When does the puck hit its maximum height? b. What is the maximum height of the puck? fez i t t 6 4 c. When does the puck hit the ground? t C I 6 t d. What is the y-intercept? What does it represent? 6 e. Graph the function. Initial height i

13 Answer Key 1. a) 1,, 4, 8, 16 b) 4, 10, 16,, 8 c) 4, 7, 10, 13, 16 d) 3, 9, 7, 81, Explicit: f(n) = n Recursive: f(1) =, f(n+1) = f(n) +, n 1 7. a) yes d) no b) no e) yes c) yes f) no. a) f()=10, arithmetic, add b) f()=6, geometric, mult by c) f() = 16, geometric, exponential d) f() = -7, arithmetic, linear 5. a) -3 b) 1 c) passes the vertical line test 9. (-, ) 3. Recursive requires that you know the first term, and each additional term is based on the term before it. Explicit is just a formula and you can find any term (you do not need to know the term before it). 6. a) D= {1, 3, 5}, R={, 4,6) yes b) D= {1,, 3}, R={1} yes c) D={3, 4, 6}, R = {1,, 3, 4} no d) D = [0, ), R= [0, 100], no e) D = [0,5], R = [0, 13.5], yes U [0, ) [0, ) 17. (-3, 0) 18. (-, -3] 19. (-, 4) and (0, 1) 0. a) exponential decay b) exponential growth c) exponential decay d) linear growth e) exponential growth f) linear growth g) exponential decay h) linear growth 1. linear: f(m) = 3m. exponential growth, on the fifth day. i) exponential growth 3. after 4 years 4. (a) linear (b) linear (c) quadratic (d) exponential 5. approx. 897 deer ( ) ( ) t grams 9. d bacteria a 36. Vertical shrink /3 Reflection across the x- axis (flip) Horizontal shift right 3 Vertical shift down y = x h(x) reflect over x Horizontal shift left 1 Vertical shift up g(x) Horizontal shift right 1 Vertical shift up 3

14 (a) 41. (b) k(x) Reflect over x x -y-3 x+1 y+ Horizontal shift right Vertical shift up (x-5)(x+5) 43. (x-6)(x-4) 44. (3x-8)(x-1) 45. 3(x-7)(x+1) 46. xy(3-x+y) 47. xy ( x + 3y + 4) 48. (y+7)² 49. (k-)(3k+4) 50. x(x-9)(x-8) 51. x = 6 and - 5. a) y=(x+4)²-10 b) f(b) = (b-)² x = 4 and axis of sym: x = 7 Vertex (7, 8) 53. Vertical Stretch by Horizontal shift left 4 Vertical shift down x = -8 and 57. x = ± (a) (b) (a) Vertex:, 4 solutions because you can use the 0 product property to easily find them since the problem is already in intercept form. 60. Two x-intercepts the discriminant is positive (b) Vertex:, 4 0 solutions because the discriminant is negative 61. f(x)=(x+3)²-4 (c) ( 7) 10 y = x y = 4(x+)² Various answers Vertical stretch by Horizontal shift right 4 Vertical shift up 5

15 68. a) [0, 45] b) [0, 16] c) 0 seconds d) 16 feet e) -5 f) 6 g) (0, 45) h) 6 sec and 34 sec 69. a) 1.5 seconds b) 4 ft. c) -0.1 and 3.1 d) 6, initial height e)