Transformations of Parabolas
Assignment 2
Problem #7
By Erin Cain
This problem asks the following:
Explore the graph of y = ax2
for different values of a.
To begin with, we need to point out
that the standard equation for a parabola is the quadratic equation y = ax2 + bx +c. When we think of the simplest parabola, we
think of y = x2. In this
situation, our a = 1, b = 0, and c = 0.
When a = 1, b = 0, and c = 0, the parabola
is centered at the origin, (0, 0).
What happens when we change a?
Lets let a = 4, so we have the equation y = 4x2 . The graph is below:
It is very helpful to graph y = 4x2
on top of the graph y = x2. This allows us to easily see what
happens a varies. From the above
graph, we can see that when a increases from 1 to 4, the graph becomes
skinnier. This can be checked
easily if we think about the equation algebraically. If you multiply x2
by 4, the graph is going to increase much faster than simply x2.
Therefore, when x = 2, y = x2 will give us the point (2, 4) and y =
4x2 will give us the point (2, 16). Once again we can see that the graph will
be skinnier due to the fact that with the same input of 2, the output is larger
in the equation y = 4x2.
To verify this, letŐs graph a few more equations with a increasing:
Note that we also graphed
fractions that were greater than 1 and all of the graphs are skinnier than our
original y= x2 (the purple graph).
What will happen if a decreases to,
say, 1/2? Before graphing this, we
can think about it logically. We have already seen that the graph becomes
skinnier when a increases, so when we do the opposite, i.e. decrease it, we can
hypothesize that the graph will do the opposite, become wider. Now lets graph
it:
Our hypothesis seems to be correct,
the graph does become wider as a decreases. As previously mentioned, we can think
about this algebraically by looking at y = (1/2)x2 as x2
being divided by 2. When comparing
it to x2, y =(1/2) x2 will increase at a slower pace,
making it wider. For example, when
x = 2, y = x2 gives us the point (2, 4) and y = (1/2)x2
gives us the point (2, 2). The output of the equation y = x2 is
larger than the output of y = (1/2) x2, so y = x2 increases
faster this time, so it is the skinnier of the two. Therefore, it seems that
the graph becomes wider as a decreases. If we continue to decrease a gradually
from 1 to 0, we will continue to get the same result of the graph becoming
wider. This can be seen in the
following graphs:
Will the graph continue to become skinnier
when a is negative? Many students might
think this because we are continuing to decrease a from 1, the graph will just
get wider and wider, but will this really happen? Lets think about it for a minute. If we keep decreasing a to hit every number
after 1, will it ever stop becoming wider?
What about when a = 0? When a = 0, we have y = 0x2. Anytime something is multiplied by 0, we
get 0 for our answer. Therefore,
when a = 0, we find y = 0 which is the y-axis. Once we get past a = 0, we will be
dealing with our negative numbers.
If we think of our parabola as continually opening up, once it hits 0,
we will be looking at the graphs that are the opposite of y = ax2.
In other words, we will be looking at the graphs y = -ax2where a
includes all rational numbers greater than 0. This brings up the meaning of the
negative sign. The negative sign
represents the opposite of a value.
So in this case it is representing the opposite of y = ax2
which is the reflection of this graph over the x-axis. We can see this by
graphing a variety of graphs when a is negative.
We can now make the same generalizations
as we did before but using y = -x2 to compare the graphs to. Therefore, when a is between 0 and -1,
the graph will be wider than y = x2 and when a is less than -1, the
graph will be skinnier than y = -x2. We can see the total change in the graph
by looking at the following MOVIE
(note that here n represents a and n varies between -15and 15).