This document provides instruction on graphing quadratic functions. It covers: 1) Graphing quadratic functions by determining the axis of symmetry from the factored form, finding points where the function equals zero, and using those points to locate the vertex. 2) Finding the axis of symmetry and vertex of a quadratic function given its standard form. 3) Graphing additional points on a quadratic function by moving left and right of the vertex based on the value of a in the standard form.

1. In this session you will learn to In this lesson you will learn to
•Graph quadratic functions, •Graph quadratic functions,
•Solve quadratic equations. •Use factoring to determine
•Graph exponential functions points on the x-axis,
•Solve problems involving •Use these points to determine
exponential growth and decay. the axis of symmetry,
•Recognize and extend •Determine the vertex of the
geometric sequences. graph.
The Gateway Arch in January 2008
Picture: From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Gateway_Arch
Click to continue.

2. Quadratic Functions
y ax 2 bx c
is the general form of a quadratic function.
These functions can take many different forms.
Different forms can give us different information about the function.
An example would be the factored form you learned in the last session.
Example: y
y x2 2x 3
Factor the trinomial to the right.
y Clickx 1)( x 3)
( when factored.
Replace y with 0 and solve for x
0 ( x 1)( x 3) x
x when solved.
Click
1,3
This tells us when x=-1, y=0
and when x=3, y=0.
These are two points on the
graph of this function.
(-1,0), (3,0)
Click to continue.

3. Quadratic Functions
y ax 2 bx c
Is the general form of a quadratic function.
These functions can take many different forms.
Different forms can give us different information about the function.
An example would be the factored form you learned in the last session.
Example:
y
2
y x 2x 3
Notice, because y=0, these two points are
on the x-axis.
What is the x value half way between the
two numbers -1 and 3?
Click after you have answered.
1 3 There is a vertical line exactly x
1 half way between x=-1 & x=3
2 called the axis of symmetry of
the graph of this function. The
equation of this line is x = 1.
Click to continue.

4. Quadratic Functions
y ax 2 bx c
Is the general form of a quadratic function.
These functions can take many different forms.
Different forms can give us different information about the function.
An example would be the factored form you learned in the last session.
Example:
y
2
y x 2x 3
What is the y value of this function when x = 1?
Click after you have answered.
y x 2 2 x 3 (1)2 2(1) 3 -4
x
The point (1,-4) is a special point on the
graph of this function. This is the vertex
of the graph.
Click to continue.

5. Quadratic Functions
y ax 2 bx c
Is the general form of a quadratic function.
These functions can take many different forms.
Different forms can give us different information about the function.
An example would be the factored form you learned in the last session.
Example:
y
2
y x 2x 3
Other points on this graph can be found by
replacing x with a number to calculate y.
Calculate y if x = 0.
The point is (0, -3).
Click after you have answered.
Since the red line is the axis of symmetry,
x
there is a point on the other side of the line.
What are the coordinates of this point?
Click after you have answered.
The point is (2, -3).
These points give us a pretty good pattern.
Let’s graph two more points.
What is y when x = 4?
Click after you have answered.

6. Quadratic Functions
y ax 2 bx c
Is the general form of a quadratic function.
These functions can take many different forms.
Different forms can give us different information about the function.
An example would be the factored form you learned in the last session.
Example:
2 y
y x 2x 3
When x = 4, y = 5.
That point is (4,5).
What are the coordinates of the point
symmetrical to (4,5)?
x
Click after you have answered.
That point is (-2,5).
Connect these points with a smooth curve.
Click to see the curve.

7. y ax2 bx c
The values a,b,c gives us important information about the graph of the function.
The axis of symmetry can be found by using
x b
Example: 2a
y
What is the axis of symmetry in the
( 4)
graph of the function x 2(1) 2
y x2 4x 5 ? The axis of symmetry x
Click after you answer.
a = 1 & b = -4 is x = 2.
Click to continue.
Remember, the vertex is on the axis of symmetry.
To locate the vertex, compute y when x = 2.
What are the coordinates of the vertex?
Click after you answer.
y 22 4(2) 5 -9
The vertex is the point (2,-9).
Click to continue.

8. y ax2 bx c
The values a,b,c gives us important information about the graph of the function.
The axis of symmetry can be found by using
2 y
y x 4x 5
To graph more points, think of the vertex as the starting point.
In this function, a = 1.
To get other points, move right n steps and up a∙n2 steps. x
For the first step, n = 1. Move right 1 and up 1∙12 = 1 step.
Click to continue.
Mirror the symmetrical point to the left.
What is that point?
The point is (1, -8)
Click after you have answered..
Next, 2 steps right and 1∙22 = 4 steps up.
What point left of the axis is symmetrical to that point?
The point is (0,-5).
Click after you have answered.
After 3 steps right, how many steps are up? 1∙32 = 9.
Answer, then click.
What is the symmetrical point to the left? (-1, 0)
Answer, then click.
Draw a smooth curve to connect the points. Click to continue.

9. y ax2 bx c
The values a,b,c gives us important information about the graph of the function.
The axis of symmetry can be found by using
2 y
y x 4x 5
Using your graph paper, repeat this lesson, graph
axes of symmetry, each point and all the curves.
x
Click to end.