Parabolic explorations

By: John Vereen

LetŐs explore parabolic graphs of the form ax2 + bx + c = y. In our first picture, we will see how the variance of a affects the graphs of form ax2 + x + 2= y where b=1 and c=2. As the constant a increases in magnitude, the parabolas become steeper and skinnier. Mathematically, as x moves further away from 0 (positive or negative direction), y is increasing towards infinity. The larger our constant a grows in magnitude, the quicker the parabola increases towards infinity and negative infinity. Also, it is important to point out that when a and b are zero or when x =0, then all of the graphs share the intersection point of (0,c).

However, there is one special case for the constant a that we must explore that is quite unique. When a=0, we have a linear graph. The variable x2 is what gives the graphs of this form the parabolic shape because both negative and positive values for x give a positive output. Since our constant a is paired with the variable x2, a zero value for a eliminates the variable x2. Then, we are left with the equation y=bx + c, which produces linear graph.

Here, we have a set of graphs of the same form ax2 + bx + c = y. However, instead of b=1 and c=2, we have b=2 and c=1.  Generally, we see the same patterns with variances in a for the two sets of graphs. However, we see some interesting effects on the slope of the line when a=0. The larger the magnitude of the b value when a=0, the steeper the slope of the line. This is because, when we have a=0, this turns the b variable into the slope of a line.

Looking at this graph above, we still have parabolas of the form ax2 + bx + c = y. However, now the constant c is varying. When the constant c varies, then it appears that the parabola translates in the y direction the amount of the value of c, and distances are preserved through each shift.  Also, one important fact about graphs of this form is that, no matter the non-zero value for a or b, the y-intercept for each graph is equal to the value of c. This is because, when x=0, then the only value left in the equation for y to equal is the constant c; therefore, y=c when x=0.