Lesson: Graphing Quadratic Functions

California Mathematics Content Standard: 
Algebra I Section 21.0: Students graph quadratic functions and know that their roots are the x-intercepts.

We already know that a quadratic function is written as:

f(x) = ax² +bx + c 
 

• a and b are called "coefficients"
• c is a "constant".  
• a can never be zero
• x is always squared

When you graph a quadratic equation, you get a parabola:

Image obtained from http://www.algebra-class.com

The vertex is either at the top or bottom of the parabola, depending on whether the parabola opens up or down.

The zeroes are the points where the parabola crosses the x axis.


Let's explore the relationship between an equation and a graph.  The map below is an interactive parabola from mathwarehouse.com.  Click on the map in order to interact with it.  Watch what happens when you change different numbers!

Image obtained from mathwarehouse.com

Example 1
Let's look at a simple quadratic equation:


f(x) = x²

You can draw the graph by plugging in values for "x".  For example,
when x = 0, y = 0
when x = 1 or -1, y = 1
when x =2 or -2, y = 4

Now you have enough information to draw the graph:

Image obtained from mathisfun.com

What do you think happens to a graph when you change coefficients?  Click on the graphic below to launch the Quadratic Function Explorer.  Observe what happens when you slide the switches up or down to change the values of a, b, and c.  Ask yourself these questions:

1. Can you figure out which of a, b, or c determines whether the parabola opens up or down?
2.  Does the term have to be positive when the parabola opens up and negative when it opens down? 
3. What happens when a gets bigger or smaller?  What happens when a is negative? 

Image obtained from Math Open Reference

Easy, huh?  Now lets try a more complex formula.  We already know the general form of a quadratic equation is

f(x) = ax² +bx + c

but it can be helpful to rearrange the equation to look like this:


f(x) = a(x-h)² + k
where h = -b/2a and k = f/h

Why?


h and k are the coordinates of the vertex, which is the very highest or lowest point of the parabola.
In other words, h is the location of "x" and k is the location of "y"

Image obtained from http://www.mathisfun.com

Example 2

Now let's plot the following equation:

f(x) = 2x² - 12x + 16

 

Based on what we have learned you can identify the coefficients and constant:

a = 2     b = -12     c = 16

since we know h = -b/2a and k = f/h, we can find the values of h and k:

h = -(-12)/(2)(2) = 3

k = 2(3)² - 12(3) + 16 = -2

 

Now we have enough information to plot the graph!  What do you think it looks like?  Plot it yourself on a sheet of graph paper, then check the correct answer here to see if you are right (no cheating!).

Practice:

1. Create equations and see what the graphs look like: Enter your equation here and click "graph" to see it!
2. Graph the following equations on your own, then check your answer by clicking the solution: 
1. y = x² + 2x − 3                         Solution 
2. y = -2x² + 5x + 3                      Solution
3. y = 2x² + 2 x - 4                       Solution

Solving Quadratic Equations by Factoring

Today we are going to learn how to find a root of a quadratic equation by factoring.

In a previous lesson we learned that the standard form of a quadratic equation is:

ax² + bx + c = 0 

The roots of a quadratic equation are the solutions to the equation.  Take for example the following equation:

x² − 3x + 2

We need to find 2 numbers whose product is 2 and sum is -3.  What are those numbers?  -1 and -2.

Plug the two numbers into the following equation:

(x +  ) (x +   )


(x - 1) (x - 2)

Set each equation equal to zero to find the roots.

x - 1 = 0

x = 1

x - 2 = 0

x = 2

So, the roots are 1 and 2.

A double root occurs when the two roots are equal. Here's an example:

x² − 12x + 36

Can be factored as (x - 6) (x - 6)

If x = 6, then each factor will be 0, and therefore the quadratic will be 0.  6 is called a double root.

Practice Problems

Solve each of the following equations by factoring.  Remember to show your work!  Turn in your completed worksheet in class tomorrow.


1. x² + 7x + 12

2. x² + 3x − 10

3. x² − 3x + 2

4. 2x² + 7x + 3 

5. x² − 2x + 1

6. x² − x − 30

7. x² + 12x + 36

8. 3x² + x − 2

Need more practice?  Take this online factoring quiz