3 UNIT 4: QUADRATIC FUNCTIONS -- NO CALCULATOR

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1 Name: Algebra Final Exam Review, Part 3 UNIT 4: QUADRATIC FUNCTIONS -- NO CALCULATOR. Solve each of the following equations. Show your steps and find all solutions. a. 3x + 5x = 0 b. x + 5x - 9 = x + c. ( x - 7) + 9 = 31 d. ( x - 3 )( x + 4) = Sketch the graph for each of the following quadratic functions. Include the coordinates of the vertex, the y-intercept, and any x-intercepts. Your sketches do not have to be drawn to scale. a. f ( x) = -( x - 3) + 5 b. f ( x) = x + 10x + 9

2 Name: Algebra 4. The gym class was playing a heated game of kickball. When he was up, Matt kicked the ball as hard as he could. After t seconds, the height of the ball, in meters, is given by the function h( t) = -10t + 40t. a. Find h( 1 4) and explain the meaning in the context of this problem. b. Find the zeros of h( t) and explain the meaning of each one in the context of this problem. c. How high up does the ball go? d. At what times is the ball at a height of 30 meters? e. After 3.5 seconds, the ball reaches a fence that is 6 meters tall. Does the ball clear the fence?

3 Name: Algebra 5. For the questions that follow: Let f(x) = x + 8x 1 and g(x) = (x + 3) 4. a. Find the zero(s) of f(x). b. Use completing the square to write f(x) in vertex form. c. Find g( 1). d. Write g(x) in standard form, which looks like: g(x) = x + x Write the quadratic formula from memory. 7. The dimensions of a rectangle, given in centimeters, are x and 10 - x. a. Find the perimeter of the rectangle. (There is enough information to do this!) b. What values of x make sense in the context of this problem? Give your answer as an inequality.

4 Name: Algebra In parts c d, let A( x) be a function representing the area of the rectangle. c. Write a function formula for A( x). e. Find A( ) and explain the meaning of your answer in the context of this problem. f. Solve the equation A ( x) = 4 by guess and check.. Explain your answer(s) in the context of this problem. Do not solve f. Find the maximum value of A( x). Explain your answer(s) in the context of this problem. Do not solve by guess and check, and remember, this is a no calculator handout.) 8. Suppose f x ( ) = x -1x + 7 ( ). a. Find the value of f 1 b. Put the function f ( x) = x -1x + 7 into vertex form. Then, identify the vertex of f ( x). c. Find the zeros of f ( x).

5 Name: Algebra UNIT 4: QUADRATIC FUNCTIONS CALCULATORS ALLOWED 9. A rocket was launched off a tower; it rose for a while and then fell to the ground next to the tower. Its height (in feet) t seconds after its launch is given by the equation: h ( t) = -16t + 80t a. How tall is the tower? b. What is h (3) and what meaning does it have in the context of this problem? c. What was the highest it got and when did it reach this height? d. When did it land? e. When was its height 450 feet? 30. Solve for x algebraically. Do not use graphs or tables on your calculator. (Sorry, this review problem belongs in the no calculator section!) a. x + 3x - = 0 b. 3x + 5 = 4x + 1x 31. Let f ( x) = x -10x + 4 a. Factor f(x).

6 Name: Algebra b. Write f(x) in vertex form, and identify the vertex. UNIT 5: POLYNOMIAL FUNCTIONS -- NO CALCULATOR 3. Solve each of the following equations. Show your steps and find all solutions. a. x 3 + 7x = 15x b. ( x + 5)( x + 7x + 6) = Sketch the graph for the following polynomial function. Include the coordinates of the y-intercept and any x-intercepts (no vertex in this section). Your sketches do not have to be drawn to scale. a. f ( x) = ( x + 4) ( x - 1 ) 3 b. f ( x) = -x 3 + 5x + 6x c. f ( x) = ( x - 9x)( x -16)

7 Name: Algebra 34. For the questions that follow let h(x) = x 4 x 3 4x a. Find h(-1). b. Find the zero(s) of h(x). ( ) is the fourth degree polynomial graphed below and f (-) = Suppose f x Write an equation for f ( x)

8 Name: Algebra UNIT 5: POLYNOMIAL FUNCTIONS CALCULATORS ALLOWED 36. Solve the following equations. You may use your calculator but do not use graphs or tables on your calculator. a. x(x -5)(x - 4) = Perform the long division below. x 3 - x x - + 4x Let f(x) = x 4 3x 3 + x g(x) = x x + 8 a. Find f(x) g(x). b. Find f(x) g(x) c. Algebraically find the zeros of f(x). d. Algebraically find the zeros of g(x).

9 Name: Algebra 39. Let g ( x) = ( x + )(5x -5x -30) Put g(x) into standard form. 40. a. Perform the following long division. x 3-5x + 7 x + b. Is (x + ) a factor of (x 3 5x + 7)? Explain how you know.

10 Name: Algebra ANSWERS: 5-3 ± 97 a. x = 0 or x = - b. x = c. x = 7 ± 11 d. x = -6 or x = a. vertex: ( 3,5), y-intercept: ( 0,16); x-intercepts ( 8,0) and (-,0) b. vertex: (- 5, -16), y-intercept: ( 0,9); x-intercepts (- 1,0) and (- 9,0) 3 4 a. 9 meters b. 8 t = 0 and t = 4 The ball starts at a height of 0 and lands 4 seconds later. c. 40 meters d. At t = 1 second and at t = 3 seconds e. Yes, the ball is at a height of 17.5 meters. - 8 ± 7-4 ± a. x = (also correct: x = and x = - ± ) b. f ( x) = ( x + ) c. g (-1) = 4 d. g ( x) = x + 1x b ± b - 4ac b b - 4ac 6. (also correct: - ± ) a a a 7 a. The perimeter is 0 centimeters b. 0 < x < 10 c. A( x) = x( 10 - x) d. A ( ) = 16. When the width is centimeters, the area is 16 square centimeters. e. x = 4or x = 6. When the dimensions are 4 cm by 6 cm, the area is 4 square centimeters. f. The maximum area is 5 square centimeters. 8 a. 3 b. f ( x ) = ( x - 3) 1 ± 88-11, vertex: ( 3,-11) c. 3± 11 = 4 9. a. h (0) = 400 feet. b. h (3) = 496 ft. = height of rocket at t = 3 seconds. c. Max height = 500 ft. at.5 seconds. d. The rocket lands at sec ( =.5 (1+ 5)) e. The rocket s height is 450 ft. at t = {0.73, 4.68} seconds. 30. a. x = -3± 17 b. x = 7 5 and x = a. f (x) = (x -6)(x - 4) b. f (x) = (x - 5) -1 Vertex = (5, 1) 3 3 a. x = 0 or x = or x = -5 b. x = - 5 or x = -6 or x = a. x-intercepts: ( 1,0) and (- 4,0) (double root); y-intercept: ( 0, - 16) b. x-intercepts: (-1,0 ), ( 0,0), ( 6,0); y-intercept: ( 0,0),0,0,0 4,0 4,0. Check graph on calculator. c. y-intercept: ( 0 ). x-intercepts: ( 0 ), ( 9 ), (- ), ( ) 34 a. (- 1) = - h b. x = 0 and 35. f ( x) = 1 ( 4 x + 3 )( x - ) ( x - 6) 1± 17 x = 36 a. x = { 0, ± 5,4} 37. x x + 3, remainder = a. x 6 + x 5 + 1x 4 8x x b. x 4 3x 3 + 3x + x 8 c. x = {0, 1, } d. x = { 4, } g ( x) 5x 5x 40x a. x - 4x x+ b. No. Dividing by a factor always produces a remainder of 0. In part a, the remainder was not 0.