__Part 1: Varying values of
a__

First we will look at the graph when *a*
varies, but *b* and *c* are held constant. To better
understand how *a* affects the graph, let us look at the
following graphs for different values of *a, *when *b*
and *c* are held constant at *b*=0 and *c*=0.

As we can see from the above graph, positive
values of *a* either shrink or stretch the parabola. Notice
that for postive values of *a, *as* a* increases, the
graph shrinks but when *a* decreases the graph stretches.
When *a* is a negative number the graph is reflected over
the x-axis or it changes direction.

Now let us look us explore the graph when *a*
varies and *b* and *c* are held constant at b=c=1.

As you can see from the above graph, the different
positive values of *a* give us concave up parabolas and the
negative values of *a* give us a concave down parabola. When
a=0, the equation become y=x+1 (shown in green) and we no longer
have a parabola, but a straight line through the point (0,1).
You can also observe that all of the graphs no matter what the
value of *a* is go through the point (0,1).

__Part 2: Varying values of
b__

Now let us explore the equation for different
values of *b*. We will hold *a *and *c* constant
at a=c=0, so

When *a* and *c* are 0, we get a
linear equation, which is a line through the origin. When *b*
is positive, the slope of the line is positive and as *b*
gets larger, the line becomes steeper. When *b *is negative,
the slope of the line is negative and as *b *gets smaller
the line becomes steeper.

Now we will look at an another example, when
*b *varies and a=c=1.

Now we are back to exploring a parabola, when
a=c=1. As you can see from the graph, all of the equations go
through the same point (0,1). As the positive value of *b*
increases, the parabola shifts to the left and becomes wider.
As the negative values of *b* decrease the parabola shifts
to the right and also becomes wider.

__Part 3: Varying values of
c__

Now let us explore the equation for different
values of *c*. If we hold *a *and *b* constant
at a=b=0, then we get y=c, which is just a vertical line that
shifts up and down depending on the value of c.

Now we would expect that the value of *c*
would just shift our parabola in the positive or negative vertical
directions. Let's examine this by setting a=c=1, for different
values of *c*.

Our conclusion was right, as c increases the parabola just shifts up the vertical axis.