1. Warm up Put each of the following in slope intercept form 6x + 3y = 12
19 - y = 19x
5x + y - 2 = 12x – 3
y - 6x = 5
8x + y = 2x – 4

2. Review
To graph a line when we have slope intercept form, we can just plug in any value for x
Example: y = 4x + 6
Martha gets paid 12 dollars an hour and makes 10 dollars in tips every night. Make a table showing how much she makes for working different numbers of hours

3. 2/27/13 Quadratic Functions
I will be able to identify a quadratic function and its vertex
I will be able to find the maximum and minimum of a quadratic function

4. Quadratic Functions
A quadratic function is any function that can be written in the form f(x)= ax2 + bx + c
Remember a quadratic equation has a degree of ___
The most basic quadratic is y = x2
Lets plug in values for x and see what we get for y

5. Graph of a quadratic
y = x2 x y -4 -3 -2 -1 1 2 3 4

6. Try it X=
f(x)= x2
f(x)= 2x2 + 3x – 1 2 4 6 10

7. Graph of a quadratic
The graph of a quadratic function is called a parabola.
If a > 0, the parabola opens up
If a < 0, the parabola opens down

8. Quadratic

9. Vertex of a quadratic
The vertex refers to the highest or lowest point of a parabola
For a positive quadratic, the vertex is the lowest point.
The y-coordinate of the vertex is the minimum value of f.
For a negative quadratic the vertex is the highest point.
The y-coordinate of the vertex is the maximum value of f.

10. Example
Determine whether the following quadratic will have a maximum or a minimum
f(x) = 5x2 + 3x – 9
g(x) = x – 12x2
f(x) = 4x2 - 9x + 3
d. g(x) = -6x + x2

11. Practice f(x) = x2 + x – 6 g(x) = 5 + 4x – x2 f(x) = 2x2 - 5x + 2
d. g(x) = 7 - 6x - 2x2

12. Minimum and Maximums Enter the equation in your calculator
Look at the graph
Decide if you are finding a max or min
If you can’t see a max or a min, change your window
Press 2nd TRACE and choose option 3:minimum or 4:maximum
Use the left and right arrow keys to choose a point to the left of the min or max and a point to the right

13. Example 1
Ex 2. Identify whether f(x) = -2x2 - 4x + 1 has a maximum value or a minimum value at the vertex. Then give the approximate coordinates of the vertex.
First, graph the function:
Next, find the maximum value of the parabola (2nd, Trace):
Finally, max(-1, 3).

14. Example
f(x) = -x2 + 6x – 8  

15. Example
2. f(x) = 2x2 + 3x – 5 

16. By hand
If you don’t have a graphing calculator, the minimum or maximum can be computed using the following formula:
As long as the equation is of the form:
y = ax2 + bx + c

17. Examples
f(x) = -x2 + 6x – 8
f(x) = 2x2 + 3x – 5

18. Think about it
1. a. Give an example of a quadratic function that has a maximum value. How do you know that it has a maximum?
1. b. Give an example of a quadratic function that has a minimum value. How do you know that it has a minimum?

20. Practice Find the maximum or minimum of each quadratic
f(x) = -5x2 + x – 4
f(x) = 3x - 2x2 – 2
f(x) = – 5 - 6x2 + 9x
f(x) = -6x2 + 2x – 9
f(x) = + 3x - x2 – 7
f(x) = 4x2 + 2x – 1