Looking at the graph

by : April Kennedy

After thinking about problems that many high school classrooms spend much time on, the problem of how the graph changes as a changes seemed to be a topic that much time is spent on. We will look at how this graph changes for varying a' s and how a student could easily understand this concept using Graphing Calculator. For the graph where a varies, there is always a common point on each graph. For various a's, integer values or decimal values, the point (0,0) will always lie on the graph. In fact, (0,0) is the vertex of each graph. is a parabola extending to infinity, either opening upward from the origin if a is positive or opening downward from the origin if a is negative.

As the values of a increase from 0 to infinity, the parabola contracts toward the positive y-axis but never touches it. If the values of a decrease from 0 to negative infinity, the parabola contracts toward the x-axis but never touches the x - axis. By looking at the graph below, we can see the many ways that the graph changes as a varies. The examples on my graph are



(red) ,


By looking at the graph, we can tell that larger a values make the graph move closer to the y-axis. Smaller a values move the graph away from the y-axis. This works for non-integers as well as integers.


An exercise like this is one that is predominant in high school classrooms. Having the students examine one graph and then overlay another graph on top of that one seems to be a useful practice. Once the student has several graphs on the same coordinate axis, he can tell how the value of a changes the shape of the curve.

More explorations:

What happens when we have y = x^2 +a and a varies?

What about x^2 -a?