Graphing Quadratic, Absolute Value, and Cubic Functions

1. Understanding the shape To begin with it is very helpful to understand the shape of your function. A) A Quadratic function looks like a “U” that opens up or sometimes down. This shape is called a parabola. Here are some examples of quadratic functions. You might notice that the quadratic function can be wide or narrow. Open up or down. Look at the coefficients in front of the and form a guess to what causes the function to be wide or narrow, and what causes the function to open up or down.

B) An Absolute Value function has a “V” shape. Just like the quadratic function it can be wide or narrow and it has the potential to open up or down. Here’s some examples of absolute value functions.

This is a graph of | |

C) A cubic function has a snake like shape.

D) Before we start graphing, Do you have a good idea about each of our basic functions shapes? A Quadratic function looks like: An Absolute Value Function looks like: A Cubic Function looks like:

2) The first step in graphing a Quadratic, Absolute Value, or Cubic Function is finding the center of our graph. Lets start with graphs that are centered at x = 0. For a quadratic equation you will see a “ ” in the equation. The center of a quadratic equation is called the vertex. Label the vertex on each graph. Ex 1

Ex 2

Ex 3

3) Once you find the center of your function you find points on either side of your center. For instance if the center is at x = 0, you might plug in a value of x = -1 and x = 1 to find our points. It is important to find points around the center because this gives us an idea of how wide or narrow our function is. It is also a good idea to choose points that are equidistant from the center- this is because of the symmetry of the graphs we are looking at. Caution: Choosing x = 1 or -1 works in most cases- unless you have a fraction coefficient in front of your function. In this case choose values for x that remove fraction. For each of the examples above lets plug in x = -1 and x =1 to find points around the vertex. Label each of your points around the vertex. A table can be used to organize your points.

Ex 1

Ex 2

Ex 3

x y

0 0

x y

0 2

x y

0 2

Do you notice the symmetry of your points? Try one on your own. Ex 4 Graph

A) Vertex = (0, ) B) Points around vertex (-1, ) , (1, ) C) Graph

Graphing a cubic function or absolute value function centered at 0 is done similarly to graphing the quadratic. For a cubic function, you will see a “ and for the absolute value function a “| |”. Try graphing the following functions. Remember the steps for graphing are 1) Finding the center which in this case is at x =0. 2) Finding points around the center. 3) Using the functions basic shape to draw graph through the points you have found.

x y

Ex 5 Graph | |

A) Center: B) Points around the Center:

C) Graph:

Ex 6 Graph

A) Center: B) Points around center:

C) Graph:

x y

x y

Let’s do one where you choose different values around the center than x = -1 and x = 1.

Ex 7 Graph

| |

A) For the center we still plug in x =0 Center: (0, ) B) When we plug in points on either side of 0. We don’t use -1 and 1

in this case because of the fraction in front. We should use something that the denominator of fraction will go into. What is the smallest thing 2 goes into? So we use x =2 and x = -2 (because it is on the other side of 0). Points around center: (2, ), (-2, )

C) Graph

x y

So far you have only looked at graphs that are centered at x =0. Let’s consider graphs whose center is elsewhere. Let’s suppose the graph is centered at x = 1 For a quadratic function you will see a “( ) ” in the equation. For an absolute value function you will see a “| | in the equation. And for the cubic function a “( ) . Similarly, for graphs centered at x = 2 A quadratic would have a ( ) An absolute value would have a __________ A cubic function would have a __________ Note: Notice what happens when you plug a 2 in the expressions above? You can use this property to determine where our center is. A quadratic function contains a ( ) . Where is this function centered? x = ______ Ex 8 Determine the centers of each of the following functions. Write the center as an ordered pair.

A) ( ) Center: ( , )

B) | | Center: ( , )

C) ( ) Center: ( , )

D) Center: ( , )

E) | | Center: ( , )

F) Center: ( , )

Do you see where the center is? Can you tell when the function is centered at 0? What about elsewhere?

Ex 9 For A through C, let’s find values around the center.

A) Since the center(vertex) for ( ) is (3, 4), we are going to plug values in for x that are on either side of 3. So we choose x = 2 and x =4. So we get (2, 5) and (4, 5).

B) Since the center for | | is (-2, -3), we are going to plug in x = _____ and x =______ to find points around the center. So the points that we get are ( , ) and ( , ).

C) Since the center for ( ) is (1, 2), we are going to plug in x = _____ and x =______ to find points around the center. So the points that we get are ( , ) and ( , ).

Ex 10 Graph the following functions that are not centered at 0. A) ( )

x y

B) | |

C) ( )

x y

x y