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!!