Exploring the Graphs for

y = ax² + bx + c

By: Diana Brown

Construct graphs for the parabola

for different values of a, b, and c. (a, b, c can be any rational numbers).

Case # 1:

First lets explore the graphs if we fix a and b to equal 1 and vary the values of c for different integers.  Let c = {-2, -1, 0, 1, 2}

So our equations will be:

y = x² + x – 2

y = x² + x – 1

y = x² + x

y = x² + x + 1

y = x² + x + 2

Discoveries:

We can see here that the vertex (the lowest or highest point on the parabola) of each graph is similar.  The x value is the same in each graph but the y value varies.  Remember for a given quadratic the vertex can be computed by solving –b/2a for the x value and substituting this x value into the quadratic to find the y value of the vertex.  Since each graph has the same a and b, then the x value of the vertex for each equation is:

-(1)/(2*1) = -1/2.  Note that each equation is not the same so if we plug this x value of the vertex into each equation we will have different y values for the vertex.

Case # 2:

Second lets explore the graphs is we fix a and c to equal 1 an vary the values of b for different integers. Let b = {-3, -1, 0, 1, 3}

So our equations will be:

y = x² - 3x + 1

y = x² - x + 1

y = x² + 1

y = x² + x + 1

y = x² + 3x + 1

Discoveries:

We can see that in this case the vertex changes for each graph, but seems to be reflected about the y axis.  You can see that if b is negative (when a=1) the x value of the vertex of the graph is positive, and if b is positive (when a=1) the x value of the vertex of the graph is negative.  We can also see that each graph is going through the point (0,1) this is because as the x value of the graph is 0 and c is equal to 1 in each graph then the above graphs share the point (0,1).

Case # 3:

Third lets explore the graphs as we fix b and c to equal 1 an vary the values of a for different integers. Let a = {-2, -1, 0, 1, 2}

So our equations will be:

y = -2x² + x + 1

y = -x² + x + 1

y = x + x + 1

y = x² + x + 1

y = 2x² + x + 1

Discoveries:

Notice that if a is positive the graph opens up, and if a is negative then the graph opens down.  We also see that in comparing the red and purple graph that the purple graph seems to be “skinnier” than the red graph.  This case is also true when we compare the green and light blue graph.  The light blue graph seems to be “skinnier” than the green graph.

Lets take a look at some more graphs as a is varied. (this time lets let b and c = 0)

Let’s look at:

y = 1/4x²

y = 1/2x²

y = x²

y = 2x²

y = 4x²

Let’s also look at the following equations and their graphs:

y = -1/4x²

y = -1/2x²

y = -x²

y = -2x²

y = -4x²

As you can see from the above graphs, that if ׀ a ׀ > 1 that the graph seems to be “skinnier” than the basic graph of y = x² and if ׀ a ׀ < 1 then the graph is wider than the basic graph of y = x².