1. Learning Objectives for Section 2.3 Quadratic Functions
MAT 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.

3. MAT 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 ________________________________.

4. Graph of a Quadratic Function
MAT 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:

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

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

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

8. Quadratic Function in General Form
MAT 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.)

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

10. Vertex of a Quadratic Function
MAT 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:

11. Intercepts of a Quadratic Function
MAT 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.)

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

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

15. Quadratic Function in Vertex Form
MAT 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.

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

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

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

19. MAT 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( 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.

20. Graphical Solution to Break-Even Problem
MAT 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

21. Solution to Break-Even Problem (continued)
MAT 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.

22. Solution to Break-Even Problem (continued)
MAT 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?

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

24. MAT 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

25. Example of Quadratic Regression
MAT 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.

26. Example of Quadratic Regression (continued)
MAT 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 = x x

27. Example of Quadratic Regression (continued)
MAT 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.