For this assignment we will look at quadratic equations of the form  and vary values of a, b, and c while keeping the other two constant.

Given the equation  we will first look to fix the variables a and b and attempt to generalize a rule for how the graph behaves as we change the value of c.  The following is a set of graphs showing different values of c with fixed values of a and b:

Just as changing the value of b in the linear function causes the graph to shift vertically, the same is true for changing the value of c in a quadratic function.  That is in fact what we see here. Let us first look at the blue graph where c = 0.  Here the y-intercept of the graph is located at the origin.  As the value of c changes so does the y-intercept, causing the graph to shift vertically.  If you think about what we are doing in changing the value of c, we are either adding or subtracting a constant value to y.  This addition or subtraction will in turn shift every value of y vertically based on what value of c we are adding or subtracting to the rest of the function.

Next we will look at how our graph changes as the value of b is changed in the quadratic equation.  Below is a picture highlighting several different values of b as both a and c remain constant.

In analyzing the graphs of the six equations listed above, it should first be noted that they all have the constant -2.  Based on what we established in analyzing the changes in c, all six of the quadratic equations go through one common point, (0,-2).

After analyzing the graphs we see that the six graphs appear to shift both vertically and horizontally.  The largest value of b, the purple graph, gives us the graph that is shifted furthest left.  The smallest value of b, the light blue graph, gives us the graph that is furthest right.  By looking at the vertex of each of the six graphs it is quickly realized that the axis of symmetry for each graph is located on an integer value.  After looking more closely we see that the axis of symmetry is always -½ the value of b.  We will next see if this pattern continues as we change the values of a.

Next we will look at how the graph changes as the value of a is changed in the quadratic equation.  Below is a picture highlighting several different values of a as both b and c remain constant.

After examining the six graphs above, one of the first things that should be noted is that our assumption about the graph as the value of b changes does not appear to work in all of the situations above.  As a changes it does not hold true that the axis of symmetry is located at  -½ b.  Instead it appears that each graph has an axis of symmetry located at x = -½a.

As we continue to examine the change in a, we see that all three graphs in which a is positive have a graph which opens up while all three graphs in which a is negative has a graph that opens down.  Why would this be?  In multiplying the values of a by a negative number we are taking the outputs of y and negating them.  This is what leads to the reflected graph.

In addition to this we see that as the value of a moves further from 0, the graph continues to get thinner and thinner, it appears as though the graph is being stretched.  The opposite effect is true for the graphs as the value of a moves closer to 0.  Why does this occur?  If we look at what happens when a = ½, we are taking every  value and dividing by 2, this in turn causes every value of y to be closer to the vertex than it would be had the value of a been left at 1.  The opposite is true if we make the value of a greater than 1.