Lesson 10 Quadratic Functions and Equations Unit Test Answers

**Quadratic Functions, Equations, and Relations** **Unit Test: A** 1. Estimate the coordinates of the vertex for the graph. [Image of a parabola opening downwards] (4, 4) 2. Which choice shows one step in the process of writing the quadratic function f(x) = x^2 + 10x + 21 in vertex form by completing the square? A) f(x) = (x^2 + 10x + 5) + 21 - 10 B) f(x) = (x^2 + 10x + 10) + 21 - 10 C) f(x) = (x^2 + 10x + 20) + 21 - 20 D) f(x) = (x^2 + 10x + 25) + 21 - 25 3. On Mercury, the distance d (in feet) that an object falls in time t (in seconds) can be modeled by the function d(t) = 6t^2. How long would it take an object to fall 54 feet on Mercury? 3 seconds 4. Determine if each quadratic equation has real solutions. Choose Yes or No. A) x^2 + 3x - 2 = 0 Yes No B) x^2 + 2x = -3 Yes No C) 2x^2 - 2 = 3x Yes No D) 2x^2 - 2 = 3x Yes No 5. A hiker stands on a small hill and tosses a stone. The stone then follows a path with height h above the ground at time t. The function h(t) = -16t^2 1   + 16t + 32 models the stone's motion. How long does it take the stone to reach its maximum height? A) 0.5 second B) 1.0 second C) 1.5 seconds D) 2.0 seconds For 6-7, consider the equation for a circle, x^2 + y^2 + 6x - 4y - 3 = 0. 6. Write the equation in standard form. (x + 3)^2 + (y - 2)^2 = 16 7. What is the radius of the circle? 4 8. Simplify (2 + 2i)(3 - 5i). A) 6 + 6i B) 16 - 4i C) 6 - 14i D) -4 - 4i 9. Describe the direction of a parabola if its equation is (y + 1)^2 = -4(x + 2). Opens leftward For 10-11, use the function f(x) = -(x - 5)^2 + 9. 10. What is the range of the function? {y | y ≤ 9} 11. Where is the function decreasing? (5, ∞) 12. Use the graph to identify the attributes of the function. [Image of a parabola opening downwards] a. x-intercepts: x = -1, x = 1 b. axis of symmetry: x = -1 c. y-intercept: y = -6 d. domain: ℝ e. range: y ≤ -6 f. relative maximum or minimum: minimum y = -6 13. Cara and Ellen both take a walk through the park. Cara follows a path modeled by the equation y = x + 3. Ellen's path can be modeled by the equation y = -(x - 3)^2 + 8. Determine if each point is a point where the paths will cross. Choose Yes or No. A. (4, 7) Yes No B. (2, 5) Yes No C. (3, 6) Yes No D. (1, 4) Yes No 14. What is the equation of a parabola with a vertex at (0, 0), a focus at (5, 0), and a directrix x = -5? A. y^2 = 5x B. y^2 = 10x C. y^2 = 20x D. y^2 = 25x 15. Solve the system of equations to find the values of x, y, and z. -2x + y = 6 x + z = 8 x + y + z = 16 x = 2 y = 10 z = 6 16. Choose True or False for each statement about the function f(x) = 4(x + 1)(x - 7). A. It has intercepts at x = 1 and x = 7. True False B. Its axis of symmetry is x = -3. True False C. It has a minimum of y = -64 at the point (3, -64). True False D. Its range is [-64, ∞). True False 17. Which function has a maximum of -4? A. y = -(x + 3)^2 - 4 B. y = (x - 3)^2 + 4 C. y = x^2 - 4x + 3 D. y = -(x - 4)^2 - 1 1. Determine if each quadratic equation has real solutions. Choose Yes or No. A) 9x^2 + 7x = 6 Yes No B) 1/5x^2 + 8 = 9 Yes No C) 3(x^2 - 4) = 2x^2 + 3 Yes No D) 5 + 3x^2 = 7x^2 + 10 Yes No E) 2(3x^2 - 4) = 3(x^2 - 3) Yes No For 2-3, write the quadratic function in vertex form by completing the square. 2. f(x) = x^2 - 4x + 5 f(x) = (x - 2)^2 + 1 3. f(x) = x^2 + 6x - 8 f(x) = (x + 3)^2 - 17 4. Find an approximate quadratic function model for the graphed data by estimating the coordinates of the vertex and using one other point. Then   sketch a graph of the model.   [Image of a parabola] PA: f(x) ≈ 0.25(x - 4)^2 + 3 5. What is the center point of the circle with equation x^2 + y^2 - 4x + 2y + 2 = 0? (2, -1) 6. A raindrop drips from a window ledge of a building. The window ledge is 80 feet above the ground. How long does it take before the drop passes by another window ledge that is 16 feet above the ground?   2 seconds 7. A ball is thrown upward with an initial speed of 24 feet per second from a height of 6 feet above the ground. Choose True or False for each statement about the ball's path.   A. The ball will reach its maximum height after 0.75 seconds. True False B. The ball will reach its maximum height after 1.25 seconds. True False C. The maximum height that the ball will reach is 21 feet. True False D. The maximum height that the ball will reach is 15 feet. True False 8. Which of the following results in a number with both a real and an imaginary part? A. (4 - 5i) - (-5 + 3i) B. (4 + 6i)(4 - 6i) C. (7 - 3i) + (2 + 3i) D. (4 - 2i)(2 + 4i) 9. Which is the equation of a parabola with the focus (3, 6) and the directrix y = 2? A. (x + 3)^2 = 8(y - 6) B. (x - 3)^2 = 8(y - 4) C. (x + 3)^2 = 8(y + 4) D. (x - 3)^2 = 8(y   + 6)