Courrey J. Alexander

Construct graphs for the parabola y=ax2+bx+c for different values of a, b, and c. (a, b, c can be any rational numbers).

As the value of a increases while b and c are held constant, the parabola closes inward.  As the value of a decreases while b and c are held constant, the parabola widens.  For positive values of a, the graph opens up, and for negative values the graph opens down.  Of course, when a=0, the graph is no longer quadratic.  It becomes linear.

Here are the graphs when a=0.4 a=-0.4 a=20 a=-20, respectively.

Click here to examine the graph as a varies from -20 to 20.

It seems that as b varies while a and c are held constant, the axis of symmetry of the graph of the parabola translates to the right (upward) toward the y-axis as b approaches 0 from the left up to y=1.  Conversely, the axis of symmetry translates to the left (upward) toward the y-axis as b approaches 0 from the right up to y=1.    Axis of symmetry is x=(-b)/2a.  So, for a and c held constant at 1 and b=0, we have x=0/2(1), i.e., x=0.  If x=0, then ax2=0 and bx=0 while c=1.  Therefore y=1 or y=c.  Here we have the axis of symmetry at x=0 and y=1.

here to examine the graph as b varies from -20 to 20.

Investigating what happens when c varies as a and b remain constant, we can observe the point where the graph of the parabola crosses the y-axis.

Click here to examine the graph as c varies from -20 to 20.

Now we will examine the graphs of the following:

1)  Graph of the parabola as a and b vary while c remains constant,

2)  Graph of the parabola as b and c vary while a remains constant,

3)  Graph of the parabola as a and c vary while b remains constant,

4)  Graph of the parabola as a, b and c vary at the same time.