Christa
Marie Nathe Parabola Shifts

In
this investigation we are going to construct a series of graphs based on the
equation **y=ax ^{2}**

y= 2x^{2}+2x+3

y= 3x^{2}+2x+3

y= 4x^{2}+2x+3

It is obvious from the graph that when the value for **a
**changes, the
graph is altered. As the value for **a** increases, the parabola becomes thinner than its
predecessor. If the values for **a** are negative the parabolas are reflected over the
x-axis.

y= -1x^{2}+2x+3

y= -2x^{2}+2x+3

y= -3x^{2}+2x+3

y= -4x^{2}+2x+3

The following graphs reflects the various values of **b**, while** a **and **c** are kept constant. The graph
expands as the value of **b** increases and interestingly pulls to the left. We can expect that
if the values for **b**
were negative that the parabola would expand, but pull to the right. As we
observe, this is the case.

y=x^{2}+2x+3 y=x^{2}+2x+3 y=x^{2}+2x+3
y=x^{2}+2x+3** **y=x^{2}-2x+3
y=x^{2}-2x+3 y=x^{2}-2x+3 y=x^{2}-2x+3 ** **

** **

Now we
will look at the graphs of the equation where the value for **c **is changed, both positive and
negative. As you have observed
from the previous graphs, it would seem that the **c **value anchors the graph to a y-intercept. As we change **c**, while** a** and** b** remain constant one could
predict the shifts of the graphs.

y=x^{2}+2x-3
y=x^{2}+2x-4 y=x^{2}+2x-5 y=x^{2}+2x-6 y=x^{2}+2x+3 y=x^{2}+2x+4 y=x^{2}+2x+5
y=x^{2}+2x+6

As we
have observed in this investigation, the coefficients **a,** **b**
and **c** have
direct bearing on the graph.
Realizing these implications allows one to understand the functions of
the quadric equation.