Math Chapter 2 Review Solution

Pre-Calculus Chapter 2 Review Name: Key Hour: Use the function f(x) = 3x^2 - 1 to evaluate the following. 1. f(-3) 3(-3)^2 - 1 27 - 1 26 2. f(x - 3) 3(x - 3)^2 - 1 3(x^2 - 6x + 9) - 1 3x^2 - 18x + 27 - 1 3x^2 - 18x + 26 3. f(a + h) 3(a + h)^2 - 1 3(a^2 + 2ah + h^2) - 1 3a^2 + 6ah + 3h^2 - 1 4. (f(a + h) - f(a)) / h (3a^2 + 6ah + 3h^2 - 1 - (3a^2 - 1)) / h 6ah + 3h^2 / h 6a + 3h Find the domain of the function. Write your answer in interval notation. 5. f(x) = 3 / (x^2 + 5x - 2) (-∞, -5) U (-5, 2) U (2, ∞) 6. f(x) = √(x + 9) [-9, ∞) 7. f(x) = 2 / √(x - 2) (2, ∞) 8. f(x) = (x + 3) / (x - 1)(x + 7) (-∞, -7) U (-7, 1) U (1, ∞) 9. f(x) = √(x - 2) [2, ∞) 10. f(x) = x^2 - 3 (-∞, ∞) all reals Given: f(x) = 3x, if x < 0 f(x) = x + 1, if 0 ≤ x ≤ 2 f(x) = (x - 2)^2, if x > 2 Evaluate the following: 11. f(-3) 3(-3) = -9 12. f(0) 0 + 1 = 1 13. f(2) 2 + 1 = 3 14. f(4) (4 - 2)^2 = 2^2 = 4 Sketch the graph of the piece-wise function. 15. f(x) = {x + 2, if x < -2 {x^2, if x ≥ -2} 16. f(x) = {4, if x < -2 {x^2, if -2 ≤ x < 1 {-2x + 1, if x ≥ 1} 17. f(x) = {x + 4, if x ≤ 0 {4 - x^2, if 0 < x ≤ 2 {-x + 4, if x > 2} Describe the transformations in order that would be applied to the graph of f(x) = |x|. 18. f(x) = 1/2|x + 3| - 7 horizontal shift 3 left vertical shrink of 1/2 vertical shift of 7 down 19. f(x) = -6|3x| horizontal shrink of 1/3 reflection over the x-axis vertical stretch of 6 20. f(x) = |1/2x| + 1 horizontal stretch of 2 vertical shift up 1 A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. 1 21. f(x) = x^2; shift 4 units to the left and shift downward 1 unit. g(x) = (x + 4)^2 - 1 22. f(x) = |x|; shift to the right 1 unit, reflect over the x-axis, stretch vertically by a factor of 3. g(x) = -3|x - 1| 23. f(x) = x^2; stretch horizontally by a factor of 2, shrink vertically by a factor of 1/2, shift up 3. g(x) = 1/2(1/2x)^2 + 3 24. f(x) = √x; reflect over the y-axis, shrink horizontally by a factor of 1/3. g(x) = √(-3x) The graph of f is given. Use it to graph each of the following functions. 25. y = -f(1/2x) + 1 horizontal shrink 1/2 reflect over x-axis up 1 (FINAL) 26. y = 2f(x + 1) left 1 vertical stretch of 2 (FINAL) 27. y = -f(-x) - 3 reflect over y-axis down 3 (FINAL) Convert the quadratic to standard form, then find the following: 28. f(x) = 2x^2 + 12x + 12 y - 12 = 2(x^2 + 6x) y - 12 + 18 = 2(x^2 + 6x + 9) y + 6 = 2(x + 3)^2 y = 2(x + 3)^2 - 6 Vertex: (-3, -6) Max/Min Value: Min; -6 Domain: (-∞, ∞) Range: [-6, ∞) x-intercepts: (-3 ± √3, 0) y-intercept: (0, 12) 29. f(x) = -x^2 - 3x + 3 y - 3 = -(x^2 + 3x) y - 3 - 9/4 = -(x^2 + 3x + 9/4) y - 21/4 = -(x + 3/2)^2 y = -(x + 3/2)^2 + 21/4 Vertex: (-3/2, 21/4) Max/Min Value: Max; 21/4 Domain: (-∞, ∞) Range: (-∞, 21/4] x-intercepts: (-3 ± √21)/2 y-intercept: (0, 3) 30. Find the minimum or maximum value of f(x) = 10x^2 + 40x + 113. Identify if it is a min or a max. Minimum; 73 x = -40 / (2*10) = -2 f(-2) = 10(-2)^2 + 40(-2) + 113 = 73 31. A soft-drink vendor at a popular beach analyzes his sales records, and finds that if he sells x cans of soda in one day, his profit (in dollars) is given by   P(x) = -0.001x^2 + 3x - 1800. What is his maximum profit per day, and how many cans of soda must he sell to reach that profit? Number of Cans: 1500 Max. Profit: $450 32. A poster is 8 inches longer than it is wide. Find a function that models the area in terms of the length of one of its sides.   A(x) = x(x + 8) or x^2 + 8x