Mathematics Education

EMAT 6680, Professor Wilson


Exploration 2, Parabolas by Ursula Kirk


Fix the values for a and b, vary c. Make at least 5 graphs on the same axes as you vary c.

         Try an animation for the same range

         What is happening mathematicatically?

         Can you prove this is a translation and that the shape of the parabola does not change?


Graph1_Equations.JPG Graph1.JPG


What are the mathematics?

The graphs have all the same shape with no translations over the x-axis. However, by changing c, we obtain graphs with translations over the y-axis. As we move down from c=5 to c -5. The vertices of our graphs have shifted 5/4 units down. When c is positive, the graphs intercept the y-axis at positive values. When c is zero, the graph intercepts the y-axis at zero. When c is negative, the graphs intercept the y-axis at negative values.

The graphs have been shifted to the left by a unit of 1, since b=1. All the graphs are concave up, since a>0.

Vertex Form

Now we will convert our first equation from standard form to vertex form by completing the square.



Therefore for this first equation our vertex is


The rest of the parabolas will have the same h, but k will change by shifting down by 5/4.


Number of Solutions


Not all parabolas have solutions, when , the parabolas do not have any real solutions and their solutions are imaginary, when . The parabolas have two real solutions. At the parabola will have one single real solution.