1. Exploring Quadratic Graphs
Objective:
To graph quadratic functions.

2. Parabola
The graph of any quadratic
function.
It is a kind of
curve.

3. Where are parabolas seen in the real world?
The Arctic Poppy
Satellite Dishes
The Golden Gate Bridge
Trajectory
Headlights

4. Why is the parabola important?
Suspension Bridges use a parabolic design to evenly distribute the weight of the entire bridge to the supporting columns.

5. Why is the parabola important?
The Satellite Dish uses a parabolic shape to ensure that no matter where on the dish surface the satellite signal strikes, it is always reflected to the receiver.

6. Why is the parabola important?
A car’s Headlights, and common flashlights, use parabolic mirrors to project the light from the bulb into a tight beam, directing the light straight out from the car, or flashlight.

7. y = ax2 + bx + c Standard Form Examples

8. Minimum point is also called a vertex. y = ax2 + bx + c
Positive “a” values mean the parabola will open upwards and will have a minimum point
Minimum point is also
called a vertex.

9. Maximum point is also called a vertex. y = -ax2 + bx + c
Negative “a” values mean the parabola will open downwards and will have a maximum point
Maximum point is
also called a vertex.

10. Will the graph open up or down?

11. Standard Form
y = ax2
Examples

12. Steps 1. Draw a table and insert vertex of (0,0).
2. Choose two numbers greater than the x coordinate and two numbers less.
3. Solve for Y Graph

13. Ex. X Y
(X,Y)

14. (X,Y)
-2, 8
-1,2
0,0
1,2
2,8

15. Standard Form
y = ax2+c
Examples

16. Ex. X Y
(X,Y)

17. (X,Y)
-2, 11
-1,5
0,3
1,5
2,11

18. Check It Out! Example 2b
Graph the quadratic function.
y = –3x2 + 1
Make a table of values.
Choose values of x and
use them to find values
of y. x –2 –1 1 2 y 1 –2
–11
Graph the points. Then connect the points with a smooth curve.

19. Ex. X Y
(X,Y)

20. (X,Y)
-2,-8
-1,-2
0,0
1,-2
2,-8

21. Check It Out! Example 2a
Graph each quadratic function.
y = x2 + 2
Make a table of values.
Choose values of x and
use them to find values
of y. x –2 –1 1 2 y 2 3 6
Graph the points. Then connect the points with a smooth curve.

22. Additional Example 3A: Identifying the Direction of a Parabola
Tell whether the graph of the quadratic function opens upward or downward. Explain.
Write the function in the form
y = ax2 + bx + c by solving for y.
Add
to both sides.
Identify the value of a.
Since a > 0, the parabola opens upward.

23. Additional Example 3B: Identifying the Direction of a Parabola
Tell whether the graph of the quadratic function opens upward or downward. Explain.
y = 5x – 3x2
Write the function in the form y = ax2 + bx + c.
y = –3x2 + 5x
a = –3
Identify the value of a.
Since a < 0, the parabola opens downward.

24. Check It Out! Example 3a
Tell whether the graph of each quadratic function opens upward or downward. Explain.
f(x) = –4x2 – x + 1
f(x) = –4x2 – x + 1
Identify the value of a.
a = –4
Since a < 0 the parabola opens downward.

25. Lesson Quiz: Part I
1. Without graphing, tell whether (3, 12) is on the graph of y = 2x2 – 5.
2. Graph y = 1.5x2. no

26. Lesson Quiz: Part II
Use the graph for Problems 3-5.
3. Identify the vertex.
4. Does the function have a minimum or maximum? What is it?
5. Find the domain and range.
(5, –4)
maximum; –4
D: all real numbers;
R: y ≤ –4