Math 6.5 Quiz Review Solution Key

Chapter 2 Review Honors Pre-Calculus Name: Hour: Convert the quadratic to vertex (standard) form, then identify any intercepts, and sketch its graph. 1. f(x) = -4x² + 24x - 41 a. Vertex (Standard) Form: y + 41 = -4(x² - 6x + 9) y + 5 = -4(x - 3)² y = -4(x - 3)² - 5 b. Vertex: (3, -5) c. x-intercept(s): none d. y-intercept: (0, -41) [Image of a parabola opening downward, vertex at (3, -5), no x-intercepts, and y-intercept at (0, -41)] 2. f(x) = 1 - 3x - 2x² a. Vertex (Standard) Form: y - 17/8 = -2(x² + (3/2)x + (9/16)) y - 17/8 = -2(x + 3/4)² b. Vertex: (-3/4, 17/8) c. x-intercepts: 0 = -2(x + 3/4)² + 17/8 17/16 = (x + 3/4)² x + 3/4 = ±√(17/16) d. y-intercept: (0, 1) [Image of a parabola opening downward, vertex at (-3/4, 17/8), x-intercepts at -3/4 ± √(17/16), and y-intercept at (0, 1)] 3. Find the quadratic function that has a vertex at (2, 3) and passes through the point (0, 2). y - k = a(x - h)² y - 3 = a(x - 2)² 2 - 3 = a(0 - 2)² -1 = 4a a = -1/4 y - 3 = -(1/4)(x - 2)² 4. A gardener has 1500 ft of fencing to enclose three adjacent rectangular gardens. Determine the dimensions that will produce a maximum enclosed area. 1   [Image of a diagram showing three adjacent rectangular gardens with dimensions x and 750-2x] A = 750x - 2x² A = -(2x² - 750x) A = -2(x² - 375x) A = -2(x² - 375x + (375/2)²) + 2(375/2)² A = -2(x - 375/2)² + 140625 Maximum area: 140625 sq ft Dimensions: 187.5' x 375' 5. The profit P (in dollars) generated by selling x units of a certain commodity is given by P(x) = -1500 + 12x - 0.0004x². a. What is the maximum profit? b. How many units must be sold to generate it? -b/2a = -12 / (-2 * 0.0004) = 15000 P(15000) = -1500 + 12(15000) - 0.0004(15000)² = $85500 6. A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along   the river. a. Find a function that models the area of the field in terms of one of its sides. A = x(2400 - 2x)   b. What are the dimensions of the field that will produce the largest area? -b/2a = -2400 / 2(-2) = 600 600' x 1200' c. What is the largest area? 720,000 ft² Use the leading coefficient test and the zeros of the polynomial to sketch each polynomial. 7. f(x) = x³(x+4)²(3-x) -3 mult 1 0 mult 3 4 mult 2 [Image of a graph with x-intercepts at -4, 0, and 3, with the graph going to negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity] 8. f(x) = (x+1)²(x-2)² -1 mult 2 2 mult 2 [Image of a graph with x-intercepts at -1 and 2, with the graph going to positive infinity as x approaches both negative and positive infinity] 9. f(x) = -4x³ + 4x² + 15x -x(4x² - 4x - 15) -x(2x+3)(2x-5) 0 mult 1 -3/2 mult 1 5/2 mult 1 [Image of a graph with x-intercepts at -3/2, 0, and 5/2, with the graph going to positive infinity as x approaches negative infinity and negative infinity as x approaches positive infinity] 10. f(x) = x³ - 2x² - 4x + 8 x²(x-2) - 4(x-2) (x-2)²(x+2) -2 mult 1 2 mult 2 [Image of a graph with x-intercepts at -2 and 2, with the graph going to negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity] 11. f(x) = -4x³ + 24x² - 35x + 9 -4x³ + 24x² - 35x + 9 ↓ 1 2 -3 9 ↓ 1 2 -3 0 2x² - 4x - 3 (2x+1)(x-3) 0 mult 1 1/2 mult 1 3 mult 1 [Image of a graph with x-intercepts at 1/2 and 3, with the graph going to positive infinity as x approaches negative infinity and negative infinity as x approaches positive infinity] 12. f(x) = 2x³ - 4x² - 42x + 40 2x³ - 4x² - 42x + 40 ↓ 1 2 -21 20 ↓ 1 0 -21 0 2x² - 21 = 0 x² + x - 20 = 0 (x+5)(x-4) -5 mult 1 2 mult 1 4 mult 1 Find all the zeros of the function. Then write its complete factorization. 13. f(x) = 4x⁴ + 4x³ - 7x² + 4x - 2 Zeros: 1/2, -2 Factors: (2x-1)²(x+2) = f(x) 14. f(x) = x⁴ - 8x³ + 17x² - 8x + 16 Zeros: 4 mult 2, ±i Factors: (x-4)²(x-i)(x+i) = f(x) 15. f(x) = x⁴ - x³ - 2x² + x - 4 Zeros: -1, 2, (1±i√7)/2 Factors: (x+1)(x-2)(2x-1-i√7)(2x-1+i√7) = f(x) 16. f(x) = 2x⁴ - x³ - 7x² + x + 2 Zeros: 2, -1/2, (1±√5)/2 Factors: (x-2)(2x+1)(2x+1-√5)(2x+1+√5) 17. f(x) = x⁴ + 25x² + 144 Zeros: ±3i, ±4i Factors: (x-3i)(x+3i)(x-4i)(x+4i)