__How Does a Parabola Change Form? __

__by
__

__Tonya
DeGeorge
__

Given the parabola in the form: , we would like to investigate what happens to the graph as we vary one of the variables. In order to do this, we need to hold two of the variables constant so we can vary only one variable at a time.

__Varying a:__

To begin, letÕs vary *a* and set *b* and c equal to one. Therefore, the equation looks like:

Suppose we look at five different values of *a*: -2, -1, 0,
1, 2, 3 (these five values were chosen at random); what would the graph look
like? Using Graphing Calculator,
we can view all these graphs on the same coordinate plane:

What conjectures can we make? Well, if we take a look at each value, we can see that *negative*
values of *a*
make the parabola look like is it opening downward and *positive* values of *a* looks like parabola is opening
up. But what is the
difference between the graphs of the function when *a* = -2 or when *a* = -1?

Well, if we observe the graph further, we can see that when *a* = -2, the
parabola looks more ÒcompactÓ than when *a* = -1.
In other words, the ÒarmsÓ of the parabola extend out further when the
value of *a*
becomes more positive (increases).
However, it seems this does not hold true for when *a* > 0.

When we look at values 1, 2, and 3 for *a*, we can see that the ÒarmsÓ of the
parabola comes closer together as the value of *a* increases. So why do the ÒarmsÓ come closer together at one part of the
graph and go further apart on the other part of the graph as we increase the
value of *a*? It is because we have to think about
the *absolute
value* of *a*. If we consider |*a*|, we can see that if we increase the
value of *a*,
the ÒarmsÓ of the parabola come closer together. When we think about the negative values of *a*, we should
be thinking about when the values of *a* get *increasingly negative*. To clarify, if *a* = -2 and *a* = -3, we can see that the ÒarmsÓ of
the parabola are closer together when *a* = -3 (because -3 is more negative than
-2). We can see this in the graph
below:

And when *a* = 0, the parabola turns into a line. (This is because when *a* = 0, the term is
eliminated and we are left with the equation: , where, in this particular case, the slope and *y*-intercept
are both one.)

__Conclusions about varying the value of a:__

¯
When *a* < 0, the parabola is concave down. As the value of *a* approaches zero (*a* becomes more positive), the ÒarmsÓ of
the parabola open up, moving further away from each other.

¯
When *a* = 0, the quadratic function becomes a linear
function, with a slope of *b* and y-intercept of *c*.
(Since we started with the form: )

¯
When *a* > 0, the parabola is concave up. As the value of *a* increases, the ÓarmsÓ of the parabola
come closer together.

When we move from negative values of *a* to positive values of *a*, the graph
starts concave down, opens up and eventually becomes a line, and then flips to
concave up, where the ÒarmsÓ of the parabola come closer together. We can see this in the animation below:

__
__

__Varying b:__

Given this form: , we want to see what happens to the function when we vary *b*.

So, letÕs vary *b* by keeping *a* and *c* equal to one:

Again, letÕs look at five different values of *b*: -2, -1, 0,
1, 2, 3 (just to keep numbers consistent) and observe the graph below:

It is interesting to see that all the parabolas intersect at the same point: (0,1). Does this have any significance?

Well, letÕs look at the animation below:

It looks as though the parabola is moving from right to left
as *b*
becomes more negative. However, it
seems to move around the point (0,1).
Could this be because we set the value of *c* to 1? Just to check and see if the graph changes, letÕs change the
value of *c*
to 3. If we do that, we get:

Using the same set of values we have used for the value of *b* before
{-2,-1,0,1,2,3}, we get the following graph:

And animation:

We see the same thing happening in these graphs as well,
except the function moves around the point (0,3). Since we have not investigated the value of *c* yet
(although we have a pretty good idea what it does), the only thing we can say
about the value of *b* is that it moves the function left or right, depending on the
sign of the value of *b*. For example, when *b* is
negative, we see the graph of the function is moved over to the right of the *y*-axis and as
*b* gets
increasingly more negative, the parabola is shifted downward (into the fourth
quadrant). When *b *is
positive, it moves the function to the left of the *y*-axis and as the value of *b* increases,
the parabola is shifted downward (into the third quadrant). And finally, when *b* =0, the parabola has a minimum point
at (0,*c*)
and is directly centered on the coordinate plane. Thus, we can see that the minimum value never shifts
up above the value of *c.*

__Conclusions about varying the value of b:__

¯
When *b* < 0, the parabola is moved to the right of
the *y*-axis. As the value of *b* becomes more negative (increases
negatively), the graph of the parabola is shifted downward into the fourth
quadrant.

¯
When *b* = 0, the parabola is centered on the *y*-axis with
itÕs minimum point at (0,*c*).

¯
When *b* > 0, the parabola is moved to the left of
the *y*-axis. As the value of *b* increases, the graph of the parabola
is shifted downward into the third quadrant.

__Varying c:__

Given this form: , we want to see what happens to the function when we vary *c*.

So, letÕs vary *c* by setting *a *= 1 and *b* = 0:

Again, letÕs look at five different values of *c*: -2, -1, 0,
1, 2, 3 (keeping numbers consistent) and observe the graph below:

As weÕve hinted in the previous investigation (of *b*), we now
see that the value of *c* can shift the parabola up or down depending on *c*. We can see this in the animation as
well, shown below:

But where exactly does it shift the parabola?

Well, if we look at positive values of c, we can see that
the graph is shifted upward and that the minimum point is always at (0,*c*). When *c* is negative, the parabola is shifted
downward and the minimum point is still at (0,*c*) – but in this case *c* is negative. And when *c* = 0, the parabola has itÕs minimum
point at (0,0).

__Conclusions about varying the value of c:__

¯
When *c* < 0, the parabola is shifted downward *c* units.

¯
When *c* = 0, the parabola is not shifted in either
direction and touches the *x*-axis.

¯
When *c* > 0, the parabola is shifted upward *c* units.

__Final Conclusion:__

From this investigation, we have seen the quadratic function change depending on different values of a, b, and c. LetÕs put all this information together:

For the given function:

__When changing the value of a:__

¯
When *a* < 0, the parabola is concave down. As the value of *a* approaches zero (*a* becomes more positive), the ÒarmsÓ of
the parabola open up, moving further away from each other.

¯
When *a* = 0, the quadratic function becomes a linear
function, with a slope of *b* and y-intercept of *c*.
(Since we started with the form: )

¯
When *a* > 0, the parabola is concave up. As the value of *a* increases, the ÓarmsÓ of the parabola
come closer together.

__When changing the value of b:__

¯
When *b* < 0, the parabola is moved to the right of
the *y*-axis. As the value of *b* becomes more negative (increases
negatively), the graph of the parabola is shifted downward into the fourth
quadrant.

¯
When *b* = 0, the parabola is centered on the *y*-axis with
itÕs minimum point at (0,*c*).

¯
When *b* > 0, the parabola is moved to the left of
the *y*-axis. As the value of *b* increases, the graph of the parabola
is shifted downward into the third quadrant.

__When changing the value of c:__

¯
When *c* < 0, the parabola is shifted downward *c* units.

¯
When *c* = 0, the parabola is not shifted in either
direction and touches the *x*-axis.

¯
When *c* > 0, the parabola is shifted upward *c* units.