1. 1 Introduction to Chapter 5 Chapter 5 – Quadratic Functions 1. Four ways to solve them 2. How to graph quadratic functions and inequalities Remember! Bring your graphing calculators.

2. 2 What you will learn  Vocabulary!  What a quadratic function is  How to graph quadratic functions in  Standard form  Vertex form  Intercept form

3. Objective: 5.1 Graphing Quadratic Functions 3 Oh Boy! Vocabulary Quadratic Function - has the form: y = ax 2 + bx + c where a is not zero (in other words, there has to be a “squared” term). The graph of a quadratic function is ‘u’ shaped. It is called a parabola.

4. Objective: 5.1 Graphing Quadratic Functions 4 A What? Vertex: the lowest or highest point (if the parabola is upside down). Axis of symmetry: the vertical line through the vertex (basically splits the parabola in half).

5. Objective: 5.1 Graphing Quadratic Functions 5 Oooooh. Let’s graph some!  On your calculator –  Graph y = x 2  Graph y = -x 2 What happens?  Graph y = ½ x 2 How does that differ from y = x 2 ?  What can you summarize about the affect of the “a” term?

6. Objective: 5.1 Graphing Quadratic Functions 6 Graphing Without the Calculator  There are three forms of a quadratic equation that we will use for graphing:  Standard form: y = ax 2 + bx + c  Vertex form: y = a(x – h) 2 + k  Intercept form: y = a(x – p)(x – q)

7. Objective: 5.1 Graphing Quadratic Functions 7 Without The Calculator – Standard Form  Graph y = 2x 2 – 8x + 6.  x-coordinate of vertex is at  Plug in x to get the y coordinate of vertex.  Get two more points on each side of the vertex by making up values for x and calculating y.  What is the axis of symmetry?

8. Objective: 5.1 Graphing Quadratic Functions 8 You Try!  Graph y = -x 2 + x + 12  Use  Get y coordinate of vertex by plugging in x value.  Get two more points on either side of vertex by making up values for x.  Connect the dots.

9. Objective: 5.1 Graphing Quadratic Functions 9 Graphing a Function in Vertex Form  Graph  Generic form: y = a(x – h) 2 + k Vertex is at (h, k). Axis of symmetry at x = h Plot two points on either side of the vertex by making up values for x and calculating y.

10. Objective: 5.1 Graphing Quadratic Functions 10 You Try  Graph y=2(x – 1) 2 +3  Generic form: y = a(x – h) 2 + k Vertex is at (h, k). Plot two points on either side of the vertex by making up values for x and calculating y.

11. Objective: 5.1 Graphing Quadratic Functions 11 Graphing a Function in Intercept Form  Graph y = -(x + 2)(x – 4)  Generic form: y = a(x – p)(x – q)  X-intercepts at (p,0) and (q,0).  Axis of symmetry halfway between p and q.  Find y coordinate of vertex by plugging in x value of axis of symmetry.  Connect the dots

12. Objective: 5.1 Graphing Quadratic Functions 12 You Try  Graph y=-3(x+1)(x-5)  Generic form: y = a(x – p)(x – q)  X-intercepts at (p,0) and (q,0).  Axis of symmetry halfway between p and q.  Find y coordinate of vertex by plugging in x value of axis of symmetry.  Connect the dots

13. Objective: 5.1 Graphing Quadratic Functions 13 “Converting” to Standard Form  Write the following quadratic functions in standard form. a. y = -(x+4)(x-9)b. y = 3(x-1) 2 + 8

14. Objective: 5.1 Graphing Quadratic Functions 14 A Real World Problem  The percent of test subjects who felt comfortable at temperature x (in degrees Fahrenheit) can be modeled by: y = -3.678x 2 + 527.3x – 18.807 What temperature made the greatest number of test subjects comfortable? What percent felt comfortable?

15. Objective: 5.1 Graphing Quadratic Functions 15 Homework  Homework: page 253, 17-19 all, 20-28 even, 32, 34, 38, 40, 52