Math Section 2.5 Worksheet Key

Pre-Calculus Section 2.5 & 2.6 1. y = 2x² + 4x - 3 1a. Complete the square to put the quadratic in vertex form. y + 3 = 2(x² + 2x + 1) y + 5 = 2(x + 1)² y = 2(x + 1)² - 5 Check the vertex: x = -b/2a = -4/2(2) = -1 f(-1) = 2(-1)² + 4(-1) - 3 = 2 - 4 - 3 = -5 1b. Identify the vertex. (-1, -5) 1c. Does the quadratic have a maximum or a minimum value? minimum 1d. What is the max/min value? -5 1e. Identify the y-intercept. (0, -3) 1f. Identify the x-intercept(s). 0 = 2x² + 4x - 3 x = (-4 ± √40) / 4 = (-2 ± √10) / 2 ≈ 0.58, -2.58 1g. What is the domain? (-∞, ∞) 1h. What is the range? [-5, ∞) 1i. Sketch the graph [Image of the graph of y = 2x² + 4x - 3] 2. y = -x² - 4x + 12 2a. Complete the square to put the quadratic in vertex form. y - 12 = -x² - 4x y - 12 + 4 = -(x² + 4x + 4) y - 16 = -(x + 2)² y = -(x + 2)² + 16 Check the vertex: x = -b/2a = -(-4)/2(-1) = -2 f(-2) = -(-2)² - 4(-2) + 12 = -4 + 8 + 12 = 16 2b. Identify the vertex. (-2, 16) 2c. Does the quadratic have a maximum or a minimum value? maximum 2d. What is the max/min value? 16 2e. Identify the y-intercept. (0, 12) 2f. Identify the x-intercept(s). 0 = -x² - 4x + 12 0 = x² + 4x - 12 (mult. by -1) 0 = (x + 6)(x - 2) x = -6, 2 2g. What is the domain? (-∞, ∞) 2h. What is the range? (-∞, 16] 2i. Sketch the graph [Image of the graph of y = -x² - 4x + 12] Applications 3. If a ball is thrown directly upward with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t - 16t².   What is the maximum height attained by the ball? y value of the vertex 3) 25 feet x = -b/2a = -40/2(-16) = 1.25 f(1.25) = 40(1.25) - 16(1.25)² = 25 feet 4. A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x) = 80x - 0.4x². where the revenue R(x) is measured in dollars. a. What is the maximum revenue? y value of the vertex 4a) $4000 R(100) = 80(100) - 0.4(100)² = 4000 b. How many units should be manufactured to obtain this maximum? x value of the vertex 4b) 100 units x = -b/2a = -80/2(-0.4) = 100 5. The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If n trees are planted on an acre of land, then each tree produces 900-9n   apples. So the number of apples produced per acre is A(n) = n(900-9n). How many trees should be planted per acre in order to obtain the maximum yield of apples? A(n) = 900n - 9n² x value of the vertex 5) 50 trees x = -b/2a = -900/2(-9) = 50 In these exercises, you are asked to find a function that models a real-life situation. You do not need to solve - only set up the equation.   6. A rectangular building lot is three times as long as it is wide. Find a function that models its area A(w) in terms of its width   w. A(w) = w(3w) = 3w² 7. A rectangular box has a square base. Its height is half the width of the base. Find a function that models its volume V(w) in terms of its width w.   V(w) = LWH = (w)(w)(1/2 w) = 1/2 w³ 8. A rectangle has a perimeter of 20 ft. Find a function that models its area A(x) in terms of the length x of one of its sides. A(x) = x(10-x) 9. Find a function that models the radius r of a circle in terms of its area A. A = πr² r² = A/π r = √(A/π) 10. The sum of two positive numbers is 60. Find a function that models their product P(x) in terms of x, one of the numbers. First #: x Second #: 60 - x P(x) = x(60-x) 11. A right triangle has one leg twice as long as the other. Find a function that models its perimeter P(x) in terms of the length x of the shorter leg.   x² + (2x)² = c² 5x² = c² c = √5x P(x) = 3x + x√5 12. A farmer has 2400 feet of fencing and wants to fence a rectangular field that borders a straight river. He does not need a fence along the river. Find a function that models the area of the field in terms of one of its sides.   A(x) = x(2400-2x)