Math 2.1-2.5 Quiz Review Solution

2.1-2.5 Quiz Review Honors Pre-Calculus Name: Key Find the vertex, x-intercepts and y-intercept of the function. 1. f(x) = -3x² + 9x + 5 x = -b/2a = -9 / 2(-3) = 3/2 f(3/2) = -3(3/2)² + 9(3/2) + 5 = 47/4 Vertex: (3/2, 47/4) x-intercepts: 0 = -3x² + 9x + 5 x = (9 ± √9² - 4(-3)(5)) / 2(-3) x = (9 ± √141) / -6 y-intercept: (0, 5) 2. f(x) = (x + 3/2)² + 1 Vertex: (-3/2, 1) x-intercepts: None! 0 = (x + 3/2)² + 1 -1 = (x + 3/2)² y-intercept: (0, 13/4) Convert the function to vertex form. 3. f(x) = 4x² - 4x + 21 y - 21 + 1 = 4(x² - x + 1/4) y - 20 = 4(x - 1/2)² y = 4(x - 1/2)² + 20 4. f(x) = -2x² + 16x - 31 y + 31 - 32 = -2(x² - 8x + 16) y - 1 = -2(x - 4)² y = -2(x - 4)² + 1 Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. 5. Vertex: (-4, -1); Point: (-2,   4) y = a(x - h)² + k 4 = a(-2 - (-4))² - 1 4 = 4a - 1 5 = 4a a = 5/4 y = 5/4(x + 4)² - 1 6. Vertex: (-1/4, 3/2); Point: (-2, 0) 0 = a(-2 + 1/4)² + 3/2 -3/2 = a(-7/4)² a = -24/49 y = -24/49(x + 1/4)² + 3/2 7. A textile manufacturer has daily production costs of C = 10000 - 110x + 0.45x², where C is the total cost and x is the number of units produced. Determine how many units should be produced each day to yield a minimum cost?   What is the minimum cost? x = -b/2a = 110 / 2(0.45) = 122 units C = 10000 - 110(122) + 0.45(122)² ≈ 3277.80 122 Units Produced $3277.80 Minimum Cost 8. Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum. x + (100 - x) = 100 x² + (100 - x)² 2x² - 200x + 10000 x = -b / 2a = 200 / 2(2) = 50 9. The figure below shows that when a football is kicked, the nearest defensive player is 6 feet from the point of impact with the kicker's foot. The height of the punted football,   f(x), in feet, can be modeled by f(x) = -0.01x² + 1.18x + 2, where x is the ball's horizontal distance, in feet, from the point of impact with the kicker's foot. a)   At what distance from the kicker's foot does the ball reach its maximum height? X-coordinate of vertex = -b / 2a = -1.18 / 2(-0.01) = 59 ft b) What is the maximum height reached by the ball? Y-coordinate of vertex = -0.01(59)² + 1.18(59) + 2 ≈ 36.01 ft c) How far must the nearest defensive player, who is 6 feet from the kicker, reach to block the punt? x = 6, find y -0.01(6)² + 1.18(6) + 2 ≈ 8.72 ft d) If the ball is not blocked, how far down the field will it go before hitting the ground? y = 0, x-intercept 0 = -0.01x² + 1.18x + 2 Quadratic Formula x ≈ 119.67 ft 10. A farmer has 2400 feet 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.   River | | x | | | | 2400 - 2x | | |__|__| A(x) = x(2400 - 2x) or -2x² + 2400x b) What are the dimensions of the field that would produce the maximum area? x = -b / 2a = -2400 / 2(-2) = 600 600 ft by 1200 ft c) What is the maximum area? 720,000 ft² Find all the real zeros of the polynomial function (You do not need the Rational Root Theorem for this! This is from Section 2.21)   Determine the multiplicity of each zero. Use the leading coefficient test and end behavior to sketch the graph. Don't forget to show the y-intercept! 11. f(x) = x^4 + x^3 - 9x^2 - 9x - 18 (x^2 + 2)(x^2 - 9) (x + 2)(x^2 - 9) (x + 2)(x + 3)(x - 3) Zeros: -2 (mult. 1), -3 (mult. 1), 3 (mult. 1) End behavior: x → ∞, f(x) → ∞ x → -∞, f(x) → -∞ y-intercept: (0, -18) [Image of the graph of f(x) = x^4 + x^3 - 9x^2 - 9x - 18] 12. f(x) = -x^3(x + 3)^2(2 - x) Degree is 6 L.C. is positive Zeros: 0 (mult. 3), -3 (mult. 2), 2 (mult. 1) End behavior: x → ∞, f(x) → ∞ x → -∞, f(x) → ∞ [Image of the graph of f(x) = -x^3(x + 3)^2(2 - x)] Find all the real zeros of the polynomial and state their multiplicities. (You do not need the Rational Root Theorem for this! This is from Section 2.21) 13. f(x) = x^6 - 6x^4 + 9x² 0 = x²(x^4 - 6x² + 9) 0 = x(x² - 3)² x = 0 x² = 3 x = ±√3 each mult. 2!! 14. f(x) = (5/3)x² + (8/3)x - (4/3) 0 = (1/3)(5x² + 8x - 4) 0 = 5x² + 8x - 4 0 = (5x - 2)(x + 2) x = 2/5, -2 each mult. 1 For #15-18, find a polynomial function with real coefficients that satisfies the given information. Write your answer in expanded form (do not leave it in factored form).   15. The polynomial is cubic, tangent to the x-axis at -2, and contains the points (3, 0) and (0, -2).   y = (x + 2)²(x - 3) = (x² + 4x + 4)(x - 3) = x³ - 3x² + 4x² - 12x + 7x - 12 y = x³ + x² - 8x - 12 now (0, -2) must be on the graph, find "a" -2 = a(0³ + 0² - 8(0) - 12) -2 = -12a a = 1/6 y = (1/6)(x³ + x² - 8x - 12) y = 1/6x³ + 1/6x² - 4/3x - 2 16. The polynomial has zeros: -4 with multiplicity 2, and 1 ± √2. It falls to the right and left. y = (x + 4)²(x - (1 + √2))(x - (1 - √2)) = (x² + 8x + 16)(x² - 2x - 1) = x⁴ - 2x³ - x² + 8x³ - 16x² - 8x + 16x² - 32x - 16 = x⁴ + 6x³ - x² - 40x - 16 -x⁴ - 6x³ + x² + 40x + 16 17. The polynomial has zeros -1 multiplicity 2, and 2 ± 5i. (x + 2 + 5i)(x - (2 - 5i)) ((x - 2) + 5i)((x - 2) - 5i) (x - 2)² - (5i)² x² - 4x + 4 - 25i² x² - 4x + 29 y = (x + 1)²(x² - 4x + 29) = (x² + 2x + 1)(x² - 4x + 29) = x⁴ - 4x³ + 29x² + 2x³ - 8x² + 58x + x² - 4x + 29 = x⁴ - 2x³ + 22x² + 54x + 29 18. The polynomial has zeros -1, 2, and i and contains the point (1, 8). (x + 1)(x - 2)(x - i)(x + i) (x² - x - 2)(x² + 1) x⁴ + x² - x³ - x - 2x² - 2 y = x⁴ - x³ - x² - x - 2 8 = (1 - 1 - 1 - 1 - 2)a 8 = -4a a = -2 y = -2(x⁴ - x³ - x² - x - 2) or -2x⁴ + 2x³ + 2x² + 2x + 4 19. Find the quotient q(x) and the remainder r(x) for (x⁴ - 3x² + 2) ÷ (x² - 1). ``` x² - 1 | x⁴ + 0x³ - 3x² + 0x + 2 -(x⁴ - x²) ---------------- -2x² + 2 -(-2x² + 2) -------------- 0 ``` q(x) = x² - 2 r(x) = 0 20. Given f(x) = 3x³ + 2x² - 19x + 6 a) Verify that (x + 3) is a factor of the function. Show your work below. ``` -3 | 3 2 -19 6 | 3 -7 18 ----------------- 3 -7 -2 0 ``` Remainder = 0 b) Find the remaining factors of f(x) 3x² - 7x + 2 (3x - 1)(x - 2) c) Use your results from part b to show the complete factorization of f(x) f(x) = (x + 3)(3x - 1)(x - 2) d) List all real zeros of f(x) -3, 1/3, 2 all mult. 1