Introduction

1. Introduction Quadratic equations can be written in standard form, factored form, and vertex form. While each form is equivalent, certain forms easily reveal different features of the graph of the quadratic function. In this lesson, you will learn to use the processes of factoring and completing the square to show key features of the graph of a quadratic function and determine how these key features relate to the characteristics of a real-world situation. 5.6.2: Writing Equivalent Forms of Quadratic Functions

2. Key Concepts Recall that the standard form, or general form, of a quadratic function is written as f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. The process of completing the square can be used to transform a quadratic equation from standard form to vertex form, f(x) = a(x – h)2+ k. Vertex form can be used to identify the key features of a function’s graph. 5.6.2: Writing Equivalent Forms of Quadratic Functions

3. Key Concepts, continued The vertex of a parabola is the point where the graph changes from increasing to decreasing, or vice versa. In vertex form, the extremum of the graph of a quadratic equation is easily identified using the vertex, (h, k). If a < 0, the function achieves a maximum, where k is the y-coordinate of the maximum and h is the x-coordinate of the maximum. If a > 0, the function has a minimum, where k is the y-coordinate of the minimum and h is the x-coordinate of the minimum. 5.6.2: Writing Equivalent Forms of Quadratic Functions

4. Key Concepts, continued Because the axis of symmetry goes through the vertex, the axis of symmetry is easily identified from vertex form as x = h. The process of factoring can be used to transform a quadratic equation in standard form to factored form, f(x) = a(x – r)(x – s). The zeros of a function are the x-values where the function value is 0. Setting the factored form equal to 0, 0 = a(x – r)(x – s), the zeros are easily identified as r and s. 5.6.2: Writing Equivalent Forms of Quadratic Functions

5. Key Concepts, continued As long as the coefficients of x are 1, the x-intercepts can be identified as (r, 0) and (s, 0). The axis of symmetry is easily identified from the factored form as the axis of symmetry occurs at the midpoint between the zeros. Therefore, the axis of symmetry is 5.6.2: Writing Equivalent Forms of Quadratic Functions

6. Common Errors/Misconceptions confusing the attributes of different forms making errors in completing the square or factoring 5.6.2: Writing Equivalent Forms of Quadratic Functions

7. Guided Practice Example 1 Suppose that the flight of a launched bottle rocket can be modeled by the equation y = –x2 + 6x, where y measures the rocket’s height above the ground in meters and x represents the rocket’s horizontal distance in meters from the launching spot at x = 0. How far does the bottle rocket travel in the horizontal direction from launch to landing? What is the maximum height the bottle rocket reaches? How far has the bottle rocket traveled horizontally when it reaches its maximum height? 5.6.2: Writing Equivalent Forms of Quadratic Functions

8. Guided Practice: Example 1, continued Identify the zeros of the function. In the original equation, y represents the height of the bottle rocket. At launch and landing, the height of the bottle rocket is 0. Write the original equation in factored form. Set it equal to 0 to identify the zeros of the function. 5.6.2: Writing Equivalent Forms of Quadratic Functions

9. Guided Practice: Example 1, continued y = –x2+ 6x Original equation 0 = –x2+ 6x Set the equation equal to 0. 0 = –(x2 – 6x) Factor out –1. 0 = –x(x – 6) Factor the binomial. Solve for x by setting each factor equal to 0. –x = 0 or x – 6 = 0 x = 0 or x = 6 5.6.2: Writing Equivalent Forms of Quadratic Functions

10. Guided Practice: Example 1, continued The x-intercepts are at x = 0 and x = 6. Find the distance between the two points to determine how far the bottle rocket travels in the horizontal direction. 6 – 0 = 6 The bottle rocket travels 6 meters in the horizontal direction from launch to landing. 5.6.2: Writing Equivalent Forms of Quadratic Functions

11. Guided Practice: Example 1, continued Determine the maximum height of the bottle rocket. The maximum height occurs at the vertex. Write the equation in vertex form by completing the square. 5.6.2: Writing Equivalent Forms of Quadratic Functions

12. Guided Practice: Example 1, continued 5.6.2: Writing Equivalent Forms of Quadratic Functions

13. Guided Practice: Example 1, continued The vertex form is y = –(x – 3)2 + 9. The vertex is (3, 9). The maximum value is the y-coordinate of the vertex, 9. The bottle rocket reaches a maximum height of 9 meters. 5.6.2: Writing Equivalent Forms of Quadratic Functions

14. Guided Practice: Example 1, continued Determine the horizontal distance from the launch point to the maximum height of the bottle rocket. We know that the bottle rocket is launched from the point (0, 0) and reaches a maximum height at (3, 9). Subtract the x-values of the two points to find the distance traveled horizontally. 3 – 0 = 3 5.6.2: Writing Equivalent Forms of Quadratic Functions

15. Guided Practice: Example 1, continued Another method is to take the total distance traveled horizontally from launch to landing and divide it by 2 to find the same answer. This is because the maximum value occurs halfway between the zeros of the function. The bottle rocket has traveled 3 meters horizontally when it reaches its maximum height. ✔ 5.6.2: Writing Equivalent Forms of Quadratic Functions

16. Guided Practice: Example 1, continued 5.6.2: Writing Equivalent Forms of Quadratic Functions

17. Guided Practice Example 3 A football is kicked and follows a path given by y = –0.03(x – 30)2 + 27, where y represents the height of the ball in feet and x represents the ball’s horizontal distance in feet. What is the maximum height the ball reaches? What horizontal distance maximizes the height? What are the zeros of the function? What do the zeros represent in the context of the problem? What is the total horizontal distance the ball travels? If the ball reaches a height of 20.25 feet after traveling 15 feet horizontally, will the ball make it over a 10-foot-tall goal post that is 45 feet from the kicker? 5.6.2: Writing Equivalent Forms of Quadratic Functions

18. Guided Practice: Example 3, continued Determine the maximum height of the ball. The maximum occurs at the vertex. The maximum value can be identified from the vertex form of the quadratic. The quadratic, y = –0.03(x – 30)2 + 27, is already in vertex form, f(x) = a(x – h)2 + k, where the vertex is (h, k). The vertex is (30, 27) and the maximum value is 27. The maximum height the ball reaches is 27 feet. 5.6.2: Writing Equivalent Forms of Quadratic Functions

19. Guided Practice: Example 3, continued Determine the horizontal distance of the ball when it reaches its maximum height. The x-coordinate of the vertex maximizes the quadratic. The vertex is (30, 27). The ball will have traveled 30 feet in the horizontal direction when it reaches its maximum height. 5.6.2: Writing Equivalent Forms of Quadratic Functions

20. Guided Practice: Example 3, continued Determine the zeros of the function. The zeros of the function occur when the function value is 0. The factored form of the quadratic equation can be used to identify the zeros of the function. First write the function in standard form. 5.6.2: Writing Equivalent Forms of Quadratic Functions

21. Guided Practice: Example 3, continued 5.6.2: Writing Equivalent Forms of Quadratic Functions

22. Guided Practice: Example 3, continued Factor the quadratic and set it equal to 0. The zeros are 0 and 60. 5.6.2: Writing Equivalent Forms of Quadratic Functions

23. Guided Practice: Example 3, continued Determine what the zeros represent in the context of the problem. The zeros represent where the ball was kicked from at a horizontal distance of 0 feet, and where the ball lands at a horizontal distance of 60 feet. 5.6.2: Writing Equivalent Forms of Quadratic Functions

24. Guided Practice: Example 3, continued Determine the total horizontal distance the ball travels. The distance between the two zeros gives the total horizontal distance traveled, 60 feet. 5.6.2: Writing Equivalent Forms of Quadratic Functions

25. Guided Practice: Example 3, continued Determine whether the ball will clear the goal post. The scenario asked if the ball would make it over a 10-foot-tall goal post that is 45 feet from the kicker if the ball reached a height of 20.25 feet after traveling 15 feet horizontally. The distance between the two zeros is 60 feet. The axis of symmetry is half the distance between the zeros at x = 30. 5.6.2: Writing Equivalent Forms of Quadratic Functions

26. Guided Practice: Example 3, continued Fifteen feet is 15 feet to the left of the axis of symmetry. The height of the ball 15 units to the right of the axis of symmetry will be the same as the height 15 units to the left of the axis of symmetry, or 20.25 feet. This is above the goal-post height of 10 feet. The ball will go over the 10-foot-tall goal post. ✔ 5.6.2: Writing Equivalent Forms of Quadratic Functions

27. Guided Practice: Example 3, continued 5.6.2: Writing Equivalent Forms of Quadratic Functions