The equation of the axis of symmetry is. Therefore, the x-coordinate of the vertex is 2.
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1 1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. Here, a = 2, b = 8, and c = 3. To find the y-intercept, substitute. f(x) = 2(0) 2 + 8(0) 3. f(x) = 3 y-intercept = 3 The equation of axis of symmetry is. The equation of the axis of symmetry is. Therefore, the x-coordinate of the vertex is 2. Graph the function. esolutions Manual - Powered by Cognero Page 1
2 2. MULTIPLE CHOICE For which equation is the axis of symmetry x = 5? A f (x) = x 2 5x + 3 B f (x) = x 2 10x + 7 C f (x) = x x 3 D f (x) = x 2 + 5x + 2 Test choice A. Here, a = 1, b = 5, and c = 3. The equation of axis of symmetry is. So, this choice is incorrect. Test choice B. Here, a = 1, b = 10, and c = 7. The equation of axis of symmetry is. Therefore, the correct answer is choice B. esolutions Manual - Powered by Cognero Page 2
3 3. Determine whether f (x) = 5 x 2 + 2x has a maximum or a minimum value. Then find this maximum or minimum value and state the domain and range of the function. For this function, a = 1, so the graph opens down and the function has a maximum value. The maximum value of the function is the y-coordinate of the vertex. The x-coordinate of the vertex is or 1. Find the y-coordinate of the vertex by evaluating the function for x = 1. The domain is all real numbers. The range is all real numbers less than or equal to the maximum value, or. 4. PHYSICAL SCIENCE From 4 feet above the ground, Maya throws a ball upward with a velocity of 18 feet per second. The height h(t) of the ball t seconds after Maya throws the ball is given by h(t) = 16t t + 4. Find the maximum height reached by the ball and the time that this height is reached. The maximum value of the function is the y-coordinate of the vertex. The x-coordinate of the vertex is or Find the y-coordinate of the vertex by evaluating the function for t = The ball reached the maximum height of feet at seconds. esolutions Manual - Powered by Cognero Page 3
4 5. Solve 3x 2 17x + 5 = 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Graph the function The x-intercepts of the graph indicate that one solution is between 0 and 1, and the other solution is between 5 and 6. Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 6. Their sum is 15, and their product is 36. Let x be a number and so the number is (15 x). Solve the quadratic equation. So, the two numbers are 3 and Their sum is 7, and their product is 15. Let x be the first number. Then 7 x is the other number. x(7 x) = 15; x 2 + 7x 15 = 0. Because the graph of the related function does not intersect the x-axis, this equation has no real solutions. Therefore, no such numbers exist. esolutions Manual - Powered by Cognero Page 4
5 8. MULTIPLE CHOICE Using the graph of the function f (x) = x 2 + 6x 7, what are the solutions to the equation x 2 + 6x 7 = 0? A 1, 6 B 1, 6 C 1, 7 D 1, 7 The x-intercepts of the graph indicate that the solutions are 7 and 1. So, the correct answer is choice D. 9. BASEBALL A baseball is hit upward with an initial velocity of 40 feet per second. Ignoring the height of the baseball player, how long does it take for the ball to fall to the ground? Use the formula h(t) = v 0 t 16t 2 where h(t) is the height of an object in feet, v 0 is the object s initial velocity in feet per second, and t is the time in seconds. Substitute h(t) = 0 and v 0 = 40 in h(t) = v 0 t 16t 2. So, the ball takes 2.5 seconds to fall to the ground. Simplify. 10. esolutions Manual - Powered by Cognero Page 5
6 esolutions Manual - Powered by Cognero Page 6
7 The impedance in one part of a series circuit is 3 + 4j ohms and the impedance in another part of the circuit is 6 7j ohms. Add these complex numbers to find the total impedance in the circuit. 17. Solve each equation by factoring. Therefore, the roots are esolutions Manual - Powered by Cognero Page 7
8 18. Therefore, the roots are Therefore, the roots are Therefore, the roots are esolutions Manual - Powered by Cognero Page 8
9 Therefore, the roots are 10i and 10i. Therefore, the roots are 6i and 6i. 23. TRIANGLES Find the dimensions of a triangle if the base is the measure of the height and the area is 12 square centimeters. Let b and h be the base and height of the triangle. The height cannot be negative, so take h = 6 cm. esolutions Manual - Powered by Cognero Page 9
10 24. PATIO Eli is putting a cement slab in his backyard. The original slab was going to have dimensions of 8 feet by 6 feet. He decided to make the slab larger by adding x feet to each side. The area of the new slab is 120 square feet. a. Write a quadratic equation that represents the area of the new slab. b. Find the new dimensions of the slab. c. What mathematical practice did you use to solve this problem. a. b. The value of x cannot be negative, so. The dimensions of the slab are 12 feet by 10 feet. c. See students' work. esolutions Manual - Powered by Cognero Page 10
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1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. 2. Determine whether f (x)
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