1 Learning Objectives for Section 2.3 Quadratic Functions You will be able to identify and define quadratic functions, equations, and inequalities. You will be able to identify and use properties of quadratic functions and their graphs. You will be able to solve applications of quadratic functions.

2 Quadratic Functions If a, b, c are real numbers with a  0, then the function is a ___________________ function, and its graph is a __________________________.

3 Graph of a Quadratic Function For each quadratic function, we will identify the axis of symmetry: vertex: y-intercept: x-intercept(s), if any:

4 Graph of a Quadratic Function For each quadratic function, we will also note the domain range

5 Two Forms of the Quadratic Function 1) General form of a quadratic function: 2) Vertex form of a quadratic function:

6 Quadratic Function in General Form For a quadratic function in general form: 1.Axis of symmetry is 2.Vertex: 3.y-intercept: Set x = 0 and solve for y. Or we can say, find f(0) 4.x-intercepts: Set f(x) = 0 and solve for x. We can use the Quadratic Formula to solve the quadratic equation.

7 The Quadratic Formula To solve equations in the form of

8 Vertex of a Quadratic Function Example: Find axis of symmetry and vertex of 1.To find the axis of symmetry: 2.To find the vertex:

9 Intercepts of a Quadratic Function Example: Find the x and y intercepts of 1) To find the y-intercept:

10 Intercepts of a Quadratic Function (continued) 2) To find the x intercepts of :

11 Graph of a Quadratic Function Now sketch the graph of :

12 Quadratic Function in Vertex Form 1.Vertex is (h, k) 2.Axis of symmetry: x = h 3.y-intercept: Set x = 0 and solve for y. Or we can say, find f(0) 4.x-intercepts: Set f(x) = 0 and solve for x. For a quadratic function in vertex form:

13 Quadratic Function in Vertex Form Example: Find vertex and axis of symmetry of Vertex: Axis of symmetry:

14 Quadratic Function in Vertex Form Example: Find the intercepts of y-intercept: x-intercepts:

15 Quadratic Function in Vertex Form Now sketch the graph of

16 Break-Even Analysis The financial department of a company that produces digital cameras has the revenue and cost functions for x million cameras are as follows: R(x) = x( x) C(x) = x. Both have domain 1 < x < 15 Break-even points are the production levels at which ________________________. Use the graphing calculator to find the break-even points to the nearest thousand cameras.

17 Graphical Solution to Break-Even Problem 1) Enter the revenue function into y1 y1= 2) Enter the cost function into y2 y2= 3) In WINDOW, change xmin=1, xmax=15, ymin= ____, and ymax=________. 4) Graph the two functions. 5) Find the intersection point(s) using CALC 5: Intersection Link to Graphing Calculator Handout

18 Solution to Break-Even Problem (continued) Here is what it looks like if we graph the cost and revenue functions on our calculators. You need to find each intersection point separately.

19 Quadratic Regression A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression to fit a linear model to the data, we would use quadratic regression on a graphing calculator to find the function of the form y = ax 2 + bx + c that best fits the data. From the STAT CALC menu, choose 5: QuadReg

20 Example of Quadratic Regression An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles.) xMileage Using quadratic regression on a graphing calculator, find the quadratic function that best fits the data.

21 Example of Quadratic Regression (continued) Enter the data in a graphing calculator and obtain the lists below. Choose quadratic regression from the statistics menu and obtain the coefficients as shown: This means that the equation that best fits the data is: y = x x