1. Quadratic Functions and Applications

2. 1. Graph a Quadratic Function Written in Vertex Form
2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
3. Find the Vertex of a Parabola by Using the Vertex Formula
4. Solve Applications Involving Quadratic Functions
5. Create Quadratic Models Using Regression

3. Graph a Quadratic Function Written in Vertex Form
The graph of the quadratic function
is a parabola.
a < 0
a > 0
The parabola opens upward for a > 0 and opens downward for a < 0

4. Graph a Quadratic Function Written in Vertex Form
The function may be written in vertex form as
The vertex is (h, k).
a < 0
a > 0
vertex (h, k)
axis of symmetry x = h

5. Example 1:
Graph the quadratic function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

6. Example 2:
Graph the quadratic function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

8. 1. Graph a Quadratic Function Written in Vertex Form
2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
3. Find the Vertex of a Parabola by Using the Vertex Formula
4. Solve Applications Involving Quadratic Functions
5. Create Quadratic Models Using Regression

9. Use the technique of completing the square to write
Example 3:
Use the technique of completing the square to write
into vertex form and identify the vertex.
Factor out the leading coefficient of the x2 term from the two terms containing x.
Complete the square within parentheses.
Remove –25 from within the parentheses along with a factor of 2.

10. Example 4:
Write in vertex form and graph the function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

11. Example 4 continued:
Write the domain and range in interval notation.

13. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
Given a quadratic function defined by

14. 1. Graph a Quadratic Function Written in Vertex Form
2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
3. Find the Vertex of a Parabola by Using the Vertex Formula
4. Solve Applications Involving Quadratic Functions
5. Create Quadratic Models Using Regression

15. Find the Vertex of a Parabola by Using the Vertex Formula
the x-coordinate of the vertex is given by
To find the y-coordinate, evaluate
The vertex is given by

16. Example 5:
Given determine the vertex by using the vertex formula and by writing in vertex form.

17. Example 5 continued:

19. 1. Graph a Quadratic Function Written in Vertex Form
2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
3. Find the Vertex of a Parabola by Using the Vertex Formula
4. Solve Applications Involving Quadratic Functions
5. Create Quadratic Models Using Regression

20. Example 6:
At the end of his career, Sherlock Holmes retired to small farm in the country and took up a hobby of beekeeping. Being of an analytical mind, he could not just watch a bee fly from flower to flower, he formulated a mathematic model for the parabolic path the bee follows from flower A to flower B.
The path is given by
where x is the distance in inches along the ground from Sherlock.

21. Example 6 continued:
a. What is the maximum height the bee reaches as it
flies from A to B?

22. Example 6 continued:
b. How far away from Sherlock are flowers A and B?

25. 1. Graph a Quadratic Function Written in Vertex Form
2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form
3. Find the Vertex of a Parabola by Using the Vertex Formula
4. Solve Applications Involving Quadratic Functions
5. Create Quadratic Models Using Regression

26. Example 7:
Use regression to find a quadratic function to model the data. Round the coefficients to 3 decimal places. x 8 10 12 18 20 23 40 55 60 65 70 75
f (x) 5 13 30 36 35 4

27. Example 7 continued:
1. Use the STAT button, then EDIT to enter the x
and y data in two lists.
Exit this screen.

28. Example 7 continued:
2. Use the STAT button, then CALC, choose
5:QuadReg.

29. Example 7 continued:
3. Hit ENTER. Then CALCULATE.
The equation is:

30. Example 7 continued:
4. To see the data and the line graphed:
Above the y = key, Turn Plot1 ON and select STATPLOT. select the type of graph.

31. Example 7 continued:
Graph

32. Example 7 continued:
5. Enter into the
equation editor and see the parabola graphed.

33. Example 7 continued: x 8 10 12 18 20 23 40 55 60 65 70 75
f (x) 5 13 30 36 35 4