Learning Objectives for Section 2.3 Quadratic Functions MAT 103 SPRING 2009 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.

MAT 103 SPRING 2009 Quadratic Functions If a, b, c are real numbers with a  0, then the function is a ____________________________ function, and its graph is a ________________________________.

Graph of a Quadratic Function MAT 103 SPRING 2009 Graph of a Quadratic Function For each quadratic function, we will identify the axis of symmetry: vertex: y-intercept: x-intercept(s), if any:

Graph of a Quadratic Function MAT 103 SPRING 2009 Graph of a Quadratic Function For each quadratic function, we will also note the domain range

Two Forms of the Quadratic Function MAT 103 SPRING 2009 Two Forms of the Quadratic Function 1) General form of a quadratic function: 2) Vertex form of a quadratic function:

Quadratic Function in General Form MAT 103 SPRING 2009 Quadratic Function in General Form For a quadratic function in general form: Axis of symmetry is Vertex:

Quadratic Function in General Form MAT 103 SPRING 2009 Quadratic Function in General Form For a quadratic function in general form: y-intercept: Set x = 0 and solve for y. Or we can say, find f(0) (Write as an ordered pair.) x-intercepts: Set f(x) = 0 and solve for x. We can factor or use the Quadratic Formula to solve the quadratic equation. (Write intercepts as ordered pairs.)

The Quadratic Formula To solve equations in the form of MAT 103 SPRING 2009 The Quadratic Formula To solve equations in the form of

Vertex of a Quadratic Function MAT 103 SPRING 2009 Vertex of a Quadratic Function Example: Find axis of symmetry and vertex of To find the axis of symmetry: To find the vertex:

Intercepts of a Quadratic Function MAT 103 SPRING 2009 Intercepts of a Quadratic Function Example: Find the x and y intercepts of 1) To find the y-intercept: (Write as an ordered pair.)

Intercepts of a Quadratic Function (continued) MAT 103 SPRING 2009 Intercepts of a Quadratic Function (continued) 2) To find the x intercepts of : (round to nearest tenth; write as ordered pairs.)

Graph of a Quadratic Function MAT 103 SPRING 2009 Graph of a Quadratic Function Now sketch the graph of :

Quadratic Function in Vertex Form MAT 103 SPRING 2009 Quadratic Function in Vertex Form For a quadratic function in vertex form: Vertex is (h , k) Axis of symmetry: x = h y-intercept: Set x = 0 and solve for y. Or we can say, find f(0) x-intercepts: Set f(x) = 0 and solve for x.

Quadratic Function in Vertex Form MAT 103 SPRING 2009 Quadratic Function in Vertex Form Example: Find vertex and axis of symmetry of Vertex: Axis of symmetry:

Quadratic Function in Vertex Form MAT 103 SPRING 2009 Quadratic Function in Vertex Form Example: Find the intercepts of y-intercept: x-intercepts:

Quadratic Function in Vertex Form MAT 103 SPRING 2009 Quadratic Function in Vertex Form Now sketch the graph of

MAT 103 SPRING 2009 Break-Even Analysis The financial department of a company that produces digital cameras has revenue (in millions of dollars) and cost functions for x million cameras as follows: R(x) = x(94.8 - 5x) C(x) = 156 + 19.7x. 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.

Graphical Solution to Break-Even Problem MAT 103 SPRING 2009 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

Solution to Break-Even Problem (continued) MAT 103 SPRING 2009 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.

Solution to Break-Even Problem (continued) MAT 103 SPRING 2009 Solution to Break-Even Problem (continued) Now, let’s graph the PROFIT function P(x) = __________________________ Where would you find the break-even points on the graph of the profit function?

Solution to Break-Even Problem (continued) MAT 103 SPRING 2009 Solution to Break-Even Problem (continued) Use the graph to find the MAXIMUM PROFIT.  4:maximum

MAT 103 SPRING 2009 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 = ax2 + bx + c that best fits the data. From the  CALC menu, choose 5: QuadReg

Example of Quadratic Regression MAT 103 SPRING 2009 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.) x Mileage 28 45 30 52 32 55 34 51 36 47 Using quadratic regression on a graphing calculator, find the quadratic function that best fits the data. Round values to 6 decimal places.

Example of Quadratic Regression (continued) MAT 103 SPRING 2009 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 = -0.517857x2 + 33.292857x- 480.942857

Example of Quadratic Regression (continued) MAT 103 SPRING 2009 Example of Quadratic Regression (continued) If appropriate, use the model to estimate the number of miles you could get from tires inflated at a) 35 psi and b) 50 psi.