1. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-1 Chapter 1 Section 8-5 Quadratic Functions, Graphs, and Models

2. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-2 Quadratic Functions, Graphs, and Models Quadratic Functions and Parabolas Graphs of Quadratic Functions Vertex of a Parabola General Graphing Guidelines A Model for Optimization

3. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-3 Quadratic Functions A function f is a quadratic function if where a, b, and c are real numbers, with

4. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-4 Graph of y x (0, 0) (1, 1)(-1, 1) (2, 4)(-2, 4) The graph is called a parabola. Every quadratic function has a graph that is a parabola. Vertex Axis (line of symmetry)

5. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-5 Example: Graphing Quadratic Functions y x y x

6. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-6 Example: Graphing Quadratic Functions (Vertical Shift) y x y x

7. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-7 Example Graphing Quadratic Functions (Horizontal Shift) y x y

8. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-8 General Principles for Graphs of Quadratic Functions 1. The graph of the quadratic function defined by is a parabola with vertex (h, k), and the vertical line x = h as axis. 2. The graph opens upward if a is positive and downward if a is negative. 3. The graph is wider than that of f (x) = x 2 if 0 1.

9. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-9 Example: Graphing Quadratic Functions (Vertical Shift) y Graph Solution

10. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-10 Example: Finding the Vertex by Completing the Square Find the vertex of the graph of We need to put the function in the form (x – h) 2 + k by completing the square. Solution The vertex is at (–1, 2).

11. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-11 Vertex Formula The vertex of the graph of has coordinates

12. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-12 Example: Vertex Formula Find the vertex of the graph of Solution The vertex is at (–1, –4).

13. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-13 General Guidelines for Graphing a Quadratic Function Step 1Decide whether the graph opens upward or downward. Step 2Find the vertex. Step 3Find the y-intercept by evaluating f (0). Step 4Find the x-intercepts, if any, by solving f (x) = 0. Step 5Complete the graph. Plot additional points as needed.

14. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-14 Example: Graph Using General Guidelines Graph Solution Step 1a = 1 > 0 so it opens upward. Step 2The vertex is at (–1, –4). Step 3f (0) = –3, so the y-intercept is (0, –3) Step 4

15. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-15 Graphing Quadratic Functions (Vertical Shift) y Step 5 The x-intercepts are (–3, 0) and (1, 0).

16. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-16 A Model for Optimization The y-value of the vertex gives the maximum or minimum value of y, while the x-value tells us where that maximum or minimum occurs. Often a model can be constructed so that y can be optimized.

17. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-17 Example: Finding a Maximum Area A farmer has 132 feet of fencing. He wants to put a fence around three sides of a rectangular plot of land, with the side of a barn forming the fourth side. Find the maximum are that he can enclose. What dimensions give this area? Solution Let x represent the width (see picture on the next slide). If we have 2 widths, that leaves 132 – 2x for the length.

18. © 2008 Pearson Addison-Wesley. All rights reserved 8-5-18 Example: Finding a Maximum Area Area Barn xx 132 - x The area: A(x) = x(132 – 2x) = –2x 2 + 132x The vertex is (33, 2178). The farmer can enclose 2178 square feet, when the width is 33 feet and the length is 132 – 2(33) = 66 feet.