Exploring Second Degree Equations

By Colleen Garrett

In this investigation I will construct graphs for the parabola y=ax2+bx+c for different values of a, b, and c.

LetŐs begin by altering a.  Click on the link below to watch y=ax2+x as a varies from -5 to 5.

Altering a

For example, Let a=1/2 and 2.  Notice that when a>1 the graph approaches infinity more rapidly and when 0<a<1 the graph approaches infinity less rapidly.  Also note that when a is negative the graph is reflected across the x-axis.

Now letŐs alter b.  For example, let b=3 and b=1/2.  You can see that this alteration of b is causing the vertex of the parabola to shift.

Click on the link below to watch y=x2+bx as b varies from -3 to 3.

Altering b.  You can see that the vertex of the parabola is moving along a parabolic curve as b changes.  LetŐs try and solve for this curve.

You know from the quadratic equation that the x-coordinate of our vertex is equal to -b/2a.  Plugging in

x=-b/2a into our original equation y=x2+bx we get the following:

y=(-b/2a) 2 + b(-b/2a)

=(b2/4)-(b2/2)

=(b2/4)-(2b2/4)

=-b2/4

=-x2

Hence, as b changes the vertex of y=x2+x moves along the curve y=-x2.

Now letŐs alter c. Click below to watch the graph as c varies from -3 to 3.

You can see that altering c changes the y coordinate of the vertex of the parabola.  If c>0 the graph moves up the y-axis, and if c<0 the graph moves down the y=axis.

For example, let c=2 and c=-3.