1. Graphs of Quadratic Functions

2. After completing this topic, you should be able to:
Find the vertex of a quadratic function.
Determine if the vertex is the maximum or minimum point of a quadratic function.
Graph a quadratic function

3. Standard Form of a Quadratic Function
Quadratic Function A quadratic function is a function that  can be written in the form
where a, b, and c are constants and 
Standard Form of a  Quadratic Function

4. When you have a quadratic function in either form, OR ,
The graph of a quadratic function is called a parabola.  It is basically a curved shape opening up or down. What does a tell us?
When you have a quadratic function in either form, 
OR ,
if a > 0, then the parabola opens up ,
if a < 0, then the parabola opens down .

5. Vertex
The vertex is the lowest or highest point (depending on direction) on the graph of a quadratic function If your quadratic function is in the form then the vertex = Basically you will find the x value of the vertex first and then just plug that value into the function to get the y or functional value of the vertex.

6. If your quadratic function is in the form , then the vertex = (h, k).

7. Axis of Symmetry
Each parabola is symmetric about a vertical line called the axis of symmetry. 
This vertical line goes through the vertex.
The next three graphs illustrate the different aspects of the graph of a quadratic function or parabola

8. The following is the graph of the function
Note a few things about
this graph:
There is one y-intercept but no x-intercept. 
The quadratic function can have no, one or two
x-intercepts
The vertex is the lowest point on the graph. 
It is either going to be the lowest or highest point on the graph of a quadratic function
The axis of symmetry. 
It is not actually part of the graph itself, but is important in that the parabola creates a mirrored image about it. 
Note how it is symmetric about the axis of symmetry. 
Also, note how it goes through the vertex. 

9. The following is the graph of the function
Note a few things about
this graph:
There is one y-intercept and one x-intercept. 
The quadratic function can have no, one or two x-intercepts

10. The following is the graph of the function
Note a few things about this graph:
There is one y-intercept and two x-intercepts. 
The quadratic function can have no, one or two x-intercepts
The axis of symmetry. 
It is not actually part of the graph itself, but is important in that the parabola creates a mirrored image about it.  
Note how it is symmetric about the axis of symmetry. 
Also, note how it goes through the vertex.
The vertex is the highest point on the graph. 
It is either going to be the lowest or highest point on the graph of a quadratic function

11. Graphing a Quadratic Function
Step 1: Does the graph curve up or down?
Step 2: Find the vertex
Step 3: Find the intercepts.
y-intercept Reminder that the y-intercept is always where the graph crosses the y-axis which means x = 0
x-intercept Reminder that the x-intercept is always where the graph crosses the x-axis which means y = 0
Step 4: Graph the parabola.   Plot the points found in steps 2 and 3 and draw a curved line through them.