Assignment #2
Shifts of a Parabola
By: Elizabeth Gore

In this exploration
we will be looking at a quadratic equation in the form of
.
What we would like
to know is what happens to the graph of this equation when we
vary d.
We should find out
some interesting things about shifts of the parabola.

In the next few examples,
we look at what happens to the graph of our equation when we change
d.
First, when we subtract
positive values of d, we see that we will get a positive horizontal
shift.
The shape of the parabola
remains constant as well as the y-intercept of the vertex.
So, we can see that
we are only changing the x-intercept of the vertex.

The same principle
holds true for subtracting negative numbers (or adding positive
numbers).
We know see a negative
horizontal shift.
Notice, as in the first
example, when d=0, our parabola's vertex lies on the y-axis, because
we have had a 0 x-axis shift.

There is no difference
when we do not use whole numbers, as shown below.

So, what can we say
about d in the equation?
D controls the horizontal
shift, or the x-intercept, of the parabola, while everything else
about the parabola remains constant.

But, let's make this
a bit more interesting of an investigation.
How could we manipulate
the vertical shift of the parabola with its structure remaining
constant?
Let's look at this
equation
If we keep d constant,
perhaps at 3, we can vary K and see what happens.

In the next few examples,
we look at what happens to the graph of our equation when we change
k.
First, when we subtract
positive values of k, we see that we will get a negative, vertical
shift.
The shape of the parabola
remains constant as well as the x-intercept of the vertex.
So, we can see that
we are only changing the y-intercept of the vertex.

The same principle
holds true for subtracting negative numbers (or adding positive
numbers).
We know see a positive
vertical shift.
Notice, as in the first
example, when k=0, our parabola's vertex lies on the x-axis, because
we have had a 0 y-axis shift.

We can assume from
this exploration that the vertex of the parabola given by this
equation
can be manipulated
horizontally and vertically by the d and k values given to it.
We have also learned
how to know exactly where the vertex of a parabola given in this
equation form lies.

EXAMPLE
Where does the vertex
of this parabola lie?
If you guessed (5,2),
you are correct.
Return
to My HomePage