Parabolic explorations

By: John Vereen

LetŐs explore parabolic graphs of the
form *a*x^{2} + *b*x + *c *= y. In our first picture, we will see
how the variance of *a*
affects the graphs of form *a*x^{2}
+ 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 x^{2} 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 x^{2}, a zero value for *a *eliminates the variable x^{2}. Then, we are left with the
equation y=*b*x
+ *c, *which produces linear graph.

Here, we have a set of graphs of the
same form *a*x^{2} + *b*x + *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 *a*x^{2}
+ *b*x + *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.