1. Quadratic Functions 12-7 Warm Up Problem of the Day
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Quadratic Functions 12-7 Warm Up
Course 3
12-7
Quadratic Functions
Warm Up
Sandra is studying a bacteria colony that has a mass of 300 grams. If the mass of the colony doubles every 2 hours, what will its mass be after 20 hours?
307,200 grams

3. Quadratic Functions 12-7 Problem of the Day
Course 3
12-7
Quadratic Functions
Problem of the Day
The time in seconds t that it takes a penny to fall a certain distance d in feet can be modeled using the equation 2d 32
t = √ . How much time will it take for a penny to fall 64 feet?
2 s

4. Quadratic Functions Learn to identify and graph quadratic functions.
Course 3
12-7
Quadratic Functions
Learn to identify and graph quadratic functions.

5. Insert Lesson Title Here
Course 3
12-7
Quadratic Functions
Insert Lesson Title Here
Vocabulary
quadratic function
parabola

6. Course 3
12-7
Quadratic Functions
A quadratic function contains a variable that is squared. In the quadratic function
the y-intercept is c. The graphs of all quadratic functions have the same basic shape, called a parabola. The cross section of the large mirror in a telescope is a parabola. Because of a property of parabolas, starlight that hits the mirror is reflected toward a single point, called the focus.
f(x) = ax2 + bx + c

7. Course 3
12-7
Quadratic Functions
Additional Example 1A: Quadratic Functions of the Form f(x) = ax2 + bx + c
Create a table for each quadratic function, and use it to make a graph.
A. f(x) = x2 + 1
Plot the points and connect them with a smooth curve. x
f(x) = x2 + 1 –2 –1 1 2
(–2)2 + 1 = 5
(–1)2 + 1 = 2
(0)2 + 1 = 1
(1)2 + 1 = 2
(2)2 + 1 = 5

8. Course 3
12-7
Quadratic Functions
Additional Example 1B: Quadratic Functions of the Form f(x) = ax2 + bx + c
B. f(x) = x2 – x + 1
Plot the points and connect them with a smooth curve. x
f(x) = x2 – x + 1 –2 –1 1 2
(–2)2 – (–2) + 1 = 7
(–1)2 – (–1) + 1 = 3
(0)2 – (0) + 1 = 1
(1)2 – (1) + 1 = 1
(2)2 – (2) + 1 = 3

9. Quadratic Functions 12-7 Try This: Example 1A
Course 3
12-7
Quadratic Functions
Try This: Example 1A
Create a table for each quadratic function, and use it to make a graph.
A. f(x) = x2 – 1
Plot the points and connect them with a smooth curve. x
f(x) = x2 – 1 –2 –1 1 2
(–2)2 – 1 = 3
(–1)2 – 1 = 0
(0)2 – 1 = –1
(1)2 – 1 = 0
(2)2 – 1 = 3

10. Quadratic Functions 12-7 Try This: Example 1B B. f(x) = x2 + x + 1
Course 3
12-7
Quadratic Functions
Try This: Example 1B
B. f(x) = x2 + x + 1
Plot the points and connect them with a smooth curve. x
f(x) = x2 + x + 1 –2 –1 1 2
(–2)2 + (–2) + 1 = 3
(–1)2 + (–1) + 1 = 1
(0)2 + (0) + 1 = 1
(1)2 + (1) + 1 = 3
(2)2 + (2) + 1 = 7

11. Course 3
12-7
Quadratic Functions
You may recall that when a product ab is 0, either a must be 0 or b must be zero.
0(–20) = (0) = 0
You can use this knowledge to find intercepts of functions.

12. Quadratic Functions 12-7 The product is 0 when x = 5 or when x = 8.
Course 3
12-7
Quadratic Functions
The product is 0 when x = 5 or when x = 8.
Example: f(x) = (x – 5)(x – 8)
(5 – 5)(5 – 8) = 0
(8 – 5)(8 – 8)= 0
Some quadratic functions can be written in the form f(x) = (x – r)(x – s). Although the variable does not appear to be squared in this form, the x is multiplied by itself when the expressions in parentheses are multiplied together.

13. Course 3
12-7
Quadratic Functions
Additional Example 2A: Quadratic Functions of the Form f(x) = a(x – r)(x – s)
Create a table for the quadratic function, and use it to make a graph.
A. f(x) = (x – 2)(x + 3)
Plot the points and connect them with a smooth curve. x
f(x) = (x – 2)(x + 3) –2 –1 1 2
(–2 – 2)(–2 + 3) = –4
(–1 – 2)(–1 + 3) = –6
(0 – 2)(0 + 3) = –6
(1 – 2)(1 + 3) = –4
(2 – 2)(2 + 3) = 0
The parabola crosses the x-axis at x = 2 and x = –3.

14. Course 3
12-7
Quadratic Functions
Additional Example 2B: Quadratic Functions of the Form f(x) = a(x – r)(x – s)
Create a table for the quadratic function, and use it to make a graph.
B. f(x) = (x – 1)(x + 4)
Plot the points and connect them with a smooth curve. x
f(x) = (x – 1)(x + 4) –2 –1 1 2
(–2 – 1)(–2 + 4) = –6
(–1 – 1)(–1 + 4) = –6
(0 – 1)(0 + 4) = –4
(1 – 1)(1 + 4) = 0
(2 – 1)(2 + 4) = 6
The parabola crosses the x-axis at x = 1 and x = –4.

15. Quadratic Functions 12-7 Remember!
Course 3
12-7
Quadratic Functions
The x-intercepts are where the graph crosses the x-axis.
Remember!

16. Quadratic Functions 12-7 Try This: Example 2A
Course 3
12-7
Quadratic Functions
Try This: Example 2A
Create a table for the quadratic function, and use it to make a graph.
A. f(x) = (x – 1)(x + 1)
Plot the points and connect them with a smooth curve. x
f(x) = (x – 1)(x + 1) –2 –1 1 2
(–2 – 1)(–2 + 1) = 3
(–1 – 1)(–1 + 1) = 0
(0 – 1)(0 + 1) = –1
(1 – 1)(1 + 1) = 0
(2 – 1)(2 + 1) = 3
The parabola crosses the x-axis at x = 1 and x = –1.

17. Quadratic Functions 12-7 Try This: Example 2B
Course 3
12-7
Quadratic Functions
Try This: Example 2B
Create a table for the quadratic function, and use it to make a graph.
B. f(x) = (x – 1)(x + 2)
Plot the points and connect them with a smooth curve. x
f(x) = (x – 1)(x + 2) –2 –1 1 2
(–2 – 1)(–2 + 2) = 0
(–1 – 1)(–1 + 2) = –2
(0 – 1)(0 + 2) = –2
(1 – 1)(1 + 2) = 0
(2 – 1)(2 + 2) = 4
The parabola crosses the x-axis at x = 1 and x = –2.

18. Additional Example 3: Application
Course 3
12-7
Quadratic Functions
Additional Example 3: Application
A reflecting surface of a television antenna was formed by rotating the parabola f(x) = 0.1x2 about its axis of symmetry. If the antenna has a diameter of 4 feet, about how much higher are the sides than the center?

19. Additional Example 3 Continued
Course 3
12-7
Quadratic Functions
Additional Example 3 Continued
First, create a table of values and graph the cross section.
The center of the antenna is at x = 0 and the height is 0 ft. If the diameter of the mirror is 4 ft, the highest point on the sides at x = 2, and the height
f(2) = 0.1(2)2 = 0.4 ft. The sides are 0.4 ft higher than the center.

20. Insert Lesson Title Here
Course 3
12-7
Quadratic Functions
Insert Lesson Title Here
Lesson Quiz: Part 1
Create a table for the quadratic function, and use it to make a graph.
1. f(x) = x2 + 2x – 1 x –2 –1 1 2 y 7

21. Insert Lesson Title Here
Course 3
12-7
Quadratic Functions
Insert Lesson Title Here
Lesson Quiz: Part 2
Create a table for each quadratic function, and use it to make a graph.
2. f(x) = (x – 2)(x + 2) x –2 –1 1 2 y –3 –4

22. Insert Lesson Title Here
Course 3
12-7
Quadratic Functions
Insert Lesson Title Here
Lesson Quiz: Part 3
3. The function f(t) = 40t – 5t2 gives the height of an arrow in meters t seconds after it is shot upward. What is the height of the arrow after 5 seconds?
75 m