Exploring the Shifts of Parabolas

by

Rita Meyers

In this exploration we will look at both the vertical
and horizontal shifts of a parabola when given the equations

We will start by looking at the horizontal
shifts
First let's look at the equation

In this equation the vertex is (d,
-1)...therefore in order to shift the parabola horizontally, you
must change the values of the equation.
So, in order to get a better understanding
of how changing d would make the parabola shift, we will look
at graph when d=1,d=3, d=6, and d=8

Looking at this graph
we can see that the parabola shifts to the right by the number
of d units, when d is a positive value.
What happens when d
is a negative value?
Let's look at the following
graph when d = -1, d
= -3, d = -6, and
d = -8

Here we see that the
parabola shifts to the left by the number of d units, when d is
a negative value.
Now that we have explored
the horizontal shifts...Let's look at the vertical shifts.
In order for the parabola
to shift vertically the values of k in the following equation
must change:

Looking at this equation,
the vertex of the parabola is (d, k)...during our exploration
we will leave the value of d = 0.
Let's look at some
possible values of k and see how it affects the parabola; we'll
start by allowing
k = 1, k = 3, k
= 6,
and k = 8

After looking at the
graphs we can see that the parabola shifts upward by k units when
the value of k is positive.
Now let's look what
happens when the value of k is negative...let's explore when k = -1, k = -3,
k = -6, and
k = -8.

By exploring these
values we can see that the parabola shifts downward the number
of k units when k denotes a negative value.
Therefore, from all
this exploring we can see how the value of d affects the horizontal
shift of the parabola and how the value of k affects the vertical
shift of the parabola...something we should also mention is no
matter what the value of d or the value of k the shape of the
parabola remains the same!!

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