Algebra II Lesson 4.1 and 4.2 Review
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1 Name: Class: Date: Algebra II Lesson 4.1 and 4.2 Review 1. Graph y = 1 4 x 2. a. c. b. d. Graph. 2. y = x 2 3 a. c. b. d. 1
2 Name: 3. y = 3x 2 + x + 1 a. c. b. d. 4. y = 2x 2 + x How would you translate the graph of y = x 2 to produce the graph of y = x ? 6. Find the vertex and the axis of symmetry of the parabola. y = 3x x Does the parabola open up or down? y = 4 + 6x 2x 2 8. Does the parabola open up or down? y = 7 5x + 3x 2 2
3 Name: Performance Assessment: 9. An object is projected into the air with a path described by the quadratic function h = 16t t, where h is the height above the ground in feet and t is the time in seconds since the object started along the path. a. What is the maximum height of the object? b. At what time does the object reach its maximum height? c. Find the time it takes the object to reach two-thirds its maximum height by using a graphing calculator to graph both h = 16t t and y = 2 3 h, where h 1 1 is the maximum height found in part (a). Use the TRACE key on the calculator to find this value. d. Will the object be this high above the ground again as it comes back down? If so, at what time? Find the maximum value or minimum value for the function. 10. f( x) = 4x 2 + 6x f( x) = 2x 2 + 5x The graph of the equation y = ax 2 4x + c has vertex (2, 5). a. Explain how to use the formula for the x-coordinate of the vertex to find the value of a. b. Use the values of x and y from the vertex in the equation to find the value of c, then write the equation. 13. Graph the parabola: y = ( x 1) a. c. b. d. 3
4 Name: 14. How would you translate the graph of y = x 2 to produce the graph of y = ( x 7) 2? Write in standard form and graph. 15. y = ( x 1) a. y = x 2 2x+ 1 c. y = x 2 + 2x+ 3 b. y = x 2 2x+ 3 d. y = x 2 + 2x y = (x + 1) 2 2 4
5 Name: Writing: 17. You have studied equations of the forms y = ( x + a) 2 and y = x 2 + b, whose graphs are translations of the graph of y = x 2. Write an explanation of how the values of a and b determine the translation that occurs. 18. Writing: Explain how to obtain the graph of y = ( x + 3) 2 2 from the graph of y = x 2. Then describe the graph of y = ( x + 3) Open-ended: Find a quadratic function that has a maximum value of 3 and x = 2 as the line of symmetry for its graph. 20. Open-ended Problem: Write a quadratic equation, if possible, for a parabola that has the following intercepts. a. one x-intercept b. two x-intercepts c. three x-intercepts d. no x-intercepts e. one y-intercept f. two y-intercepts 5
6 Algebra II Lesson 4.1 and 4.2 Review Answer Section 1. ANS: C PTS: 1 DIF: Level A REF: MAL20514 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: graph quadratic function BLM: Knowledge 2. ANS: B PTS: 1 DIF: Level B REF: MAL20517 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: graph parabola standard form quadratic function BLM: Knowledge 3. ANS: C PTS: 1 DIF: Level B REF: MAL20522 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: quadratic graph BLM: Knowledge 4. ANS: PTS: 1 DIF: Level B REF: MAL20530 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: quadratic graph parabola BLM: Knowledge 5. ANS: Move the graph of y = x 2 up 10 units to get the graph of y = x PTS: 1 DIF: Level B REF: MAL20520 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: translation parabola BLM: Analysis 6. ANS: Vertex: (-2, -3); Axis: x = -2 PTS: 1 DIF: Level B REF: MAL20525 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: axis of symmetry vertex parabola BLM: Knowledge 7. ANS: Down PTS: 1 DIF: Level A REF: MAL20531 NAT: NCTM 9-12.ALG.1.c TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: parabola down up BLM: Comprehension 1
7 8. ANS: Up PTS: 1 DIF: Level A REF: MAL20532 NAT: NCTM 9-12.ALG.1.c TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: down up parabola BLM: Comprehension 9. ANS: a. 16 ft b. 1 second c. approximately 0.42 seconds (also again at 1.58 seconds - see part (d)) d. yes; approximately 1.58 seconds PTS: 1 DIF: Level B REF: MAL20534 NAT: NCTM 9-12.NOP.3.a TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: quadratic projectile motion BLM: Application 10. ANS: minimum: 0.75 PTS: 1 DIF: Level B REF: MAL21427 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: quadratic function minimum BLM: Comprehension 11. ANS: maximum: PTS: 1 DIF: Level B REF: MAL21429 TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: quadratic function maximum BLM: Comprehension 12. ANS: a. The value of the x-coordinate of the vertex is x = b 2a. In y = ax 2 4x + c, b = 4, so solve 2 = 4 for a: a 2a = 1. b. We now have y = x 2 4x + c. Substituting 2 for x and 5 for y gives 5 = 2 2 4( 2) + c. Solving for c yields c = 9. The equation is y = x 2 4x + 9. PTS: 1 DIF: Level A REF: A SR.02 NAT: NCTM 9-12.ALG.1.c NCTM 9-12.ALG.1.e TOP: Lesson 4.1 Graph Quadratic Functions in Standard Form KEY: Quadratic vertex short response BLM: Analysis 13. ANS: C PTS: 1 DIF: Level B REF: MAL20539 NAT: NCTM 9-12.ALG.1.c KEY: graph vertex form BLM: Knowledge 2
8 14. ANS: translate the graph of y = x 2 right 7 units PTS: 1 DIF: Level B REF: MAL20540 NAT: NCTM 9-12.ALG.1.c KEY: translation parabola BLM: Comprehension 15. ANS: B PTS: 1 DIF: Level B REF: MAL20553 KEY: graph parabola quadratic BLM: Knowledge 16. ANS: y = x 2 2x 3 PTS: 1 DIF: Level B REF: MAL20552 KEY: quadratic relation graph parabola BLM: Knowledge 17. ANS: Sample answer:when a > 0, the graph of y = x 2 is translated left a units. When a < 0, the graph of y = x 2 is translated right a units. When b > 0, the graph of y = x 2 is translated up b units. When b < 0, the graph of y = x 2 is translated down b units. PTS: 1 DIF: Level B REF: MAL20554 NAT: NCTM 9-12.ALG.1.c KEY: equation square variable translate BLM: Comprehension 3
9 18. ANS: Sample answer:the graph of y = ( x + 3) 2 2 can be obtained by translating the graph of y = x 2 down 3 units and then 3 units to the left. The graph is a parabola with vertex (-3, -2) that opens upward and is congruent to the graph of y = x 2. PTS: 1 DIF: Level B REF: MAL20559 NAT: NCTM 9-12.GEO.3.a NCTM 9-12.ALG.1.e KEY: graph vertex form translation BLM: Comprehension 19. ANS: Any equation of the form y = a( x + 2) where a 0; sample: y = ( x + 2) PTS: 1 DIF: Level B REF: MAL20561 KEY: quadratic function maximum axis of symmetry BLM: Comprehension 20. ANS: Answers will vary. Examples are given. a.y = x 2 + 6x + 9 b. y = x 2 + 3x + 2 c. not possible d.y = x 2 + 2x + 5 e. y 2 = x f. y 2 = x + 3 PTS: 1 DIF: Level B REF: MAL20563 KEY: quadratic equation x-intercepts BLM: Comprehension 4
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