Constant or Not So Constant:

by

Priscilla Alexander

 

This write-up is for students who are just being introduced to the graph of a parabola. The student will see how the constants in the equation have an effect on the graph of the equation.

 

A parabolic equation is one whose highest exponent is two. For example

 or

and the graph looks like

 

    

 

The graph can look like a cup or an upside down cup with it vertex sometimes at the y-axis. A vertex is the highest or lowest turning point of a parabola. A constant is the value that , , or  can take. Now let us look at what the constants do to its graph.

 

First, let us hold  and  at one. Let us give  the values (). So the equations are

 

 

graph.gif

 

-3 takes the parabola with the color of purple, -2 takes the parabola with the color of blue,

-1 takes the parabola with the color of light blue,1 takes the parabola with the color of green,

 2 takes the parabola with the color of red

3 takes the parabola with the color of black

 

From the graphs it can be seen that the constant  determines the values of the y intercept of the parabola.

 

Next let us hold  and  at one and manipulate the constant . Let the values of  be . The equations are

 

 

graph2.gif

 

 

-1 takes the parabola with the color of purple

-2 takes the parabola with the color of blue

-3 takes the parabola with the color of light blue

3 takes the parabola with the color of green

2 takes the parabola with the color of red

1 takes the parabola with the color of black

The values of  makes the graphs shift to the left or to the right. The negative values of  makes the graph shift to the right and the positive values of  makes the graph shift to the left. The values of  make the parabola’s vertex shift ½ the value of  to the appropriate direction. Also, half the value of b is the x coordinate of the vertex.

 

Thus far we see that  is the value is the y-intercept of the parabola. In addition, that ½ the value of  tells how many places to the left or to the right. So we can expect the graph of the equation  to intercept at 5, the vertex to shift to the left 1 unit, and. See graph.

 

Third, let us hold  and  at the value one and change . Let us make the values of  be

. If the value of  is negative, the graph will open down. If the value of  is positive the graph will open upward. See graph.

 

                      

a is negative                                               a is positive

 

When  is a negative number and when the value of  is negative the vertex shifts to the left and when  is positive the vertex shifts to the right. See the above graph. When  is an integer between 1 to infinity and -1 to infinity the graph shrinks. As  approaches zero the graph gets wider. See the parabolas below.

 

 

 

graph3.gif

 

1 takes the parabola with the color of purple

2 takes the parabola with the color of blue

3 takes the parabola with the color of light blue

-2 takes the parabola with the color of green

-1 takes the parabola with the color of red

-3 takes the parabola with the color of yellow

 

Thus far we have seen that  determines the y-intercept of the graph,  determines how far and which way the vertex shifts, and  determines if the graph opens up or down, and if the graph is wide or narrow. So we can expect the equation  to intercept at the y-axis at 5, open upward, the vertex to shift to the left, and to be extremely wide.

 

 

Finally, the behavior of the graph changes when ,, or  is equal to zero.  When a value is not zero, we assume it to be one.

1.  When ,, or  are all zero there is no picture on the graph.

2.  When  is zero the y-intercept of the graph is zero

 

           3.  When  is zero, the graph does not shift, it has a y-intercept at one.

 

 

        

4.  When  is zero the graph of the equation is a line with a negative or positive slope. The slope depends on .

 

 

In conclusion, we can see that the value of  determines the y-intercept. Next we saw that the values of  determined if the graph shifted to the left or to the right. Then we saw that  determined how wide or how narrow the graph would be and  determined if the graph would open downward or upward. Finally we saw that zero was a special value for

. The most special case is when the equation goes from a parabola to a line. With all of the information given we can now determine if the values of ,, or  make the graphical behavior of the graph constant or not so constant.

 

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