1. STARTERWED, OCT 1 Given the function f(x) = 3x 2 – 12x – 36, identify these key features of the graph: 1.the extrema 2.vertex 3.y-intercept 4.x-intercepts 5.sketch the graph When you finish: Graph y = (2x – 1)(x + 3) on the calculator. Use the CALC feature to find the x-intercepts. What are they? Can you see a way to look at the factors and get the same x- intercepts? 1 1.9.1: Proving the Interior Angle Sum Theory

2. Introduction Quadratic equations can be written in several forms, including standard form, vertex form, and factored 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 various forms of quadratic functions to show the key features of the graph and determine how these key features relate to the characteristics of a real-world situation. 2 2.1.2: Interpreting Various Forms of Quadratic Functions

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4. 4 1.9.1: Proving the Interior Angle Sum Theory c = -7, so y-intercept is (0, -7) Vertex is at the point (2, 1) a = -2, therefore graph opens down and has a maximum

5. 5 2.1.2: Interpreting Various Forms of Quadratic Functions

6. 6 1.9.1: Proving the Interior Angle Sum Theory Vertex is (2, 5) Vertical line x = 2 a = 3, therefore the graph opens up and has a minimum.

7. 7 1.9.1: Proving the Interior Angle Sum Theory Factored Form continued on next page

8. 8 1.9.1: Proving the Interior Angle Sum Theory

9. 9 p = -2, q = 8 so x-intercepts are at (-2, 0) and (8, 0) So axis of symmetry is the vertical line x = 3 Plug in x =3 into the equation and find y. y = -(3 + 2)(3 – 8) = -(5)(-5) = 25, therefore the vertex is (3, 25) Continue on next page

10. 10 1.9.1: Proving the Interior Angle Sum Theory Find y when x = 0. So y = -(0 + 2)(0 – 8) = -(2)(-8) = 16 So the y-intercept is (0, 16) a = -1, therefore graph opens down and has a maximum

11. Guided Practice Example 4 Suppose that the flight of a launched bottle rocket can be modeled by the function f(x) = –(x – 1)(x – 6), where f(x) measures the height above the ground in meters and x represents the horizontal distance in meters from the launching spot at x = 1. (A)How far does the bottle rocket travel in the horizontal direction from launch to landing? (B)What is the maximum height the bottle rocket reaches? (C)How far has the bottle rocket traveled horizontally when it reaches its maximum height? (D) Graph the function. 11 2.1.2: Interpreting Various Forms of Quadratic Functions

12. Guided Practice Example 5 A football is kicked and follows a path given by f(x) = –0.03x 2 + 1.8x, where f(x) represents the height of the ball in feet and x represents the horizontal distance in feet. (A)What is the maximum height the ball reaches? (B)What horizontal distance maximizes the height? (C) Graph the function. 12 2.1.2: Interpreting Various Forms of Quadratic Functions