Assignment 2 Problem 8 for Kanita DuCloux

I will investigate several graphs of the parabola of the form

on the same axes using different values for d. I will observe the effects varying d has on the shape and position of the graph.

The general form of the parabola is

,

with parameters a, h, and k. Where the vertex is (h, k) and the axis of symmetry is x = h. In my example, I will let h = d.

I'll begin by observing the graph

where a =1, d = 0, and k = -2.

Notice that the vertex is ( d, k) or (0, -2) and the axis of symmetry is

x = d or x = 0 in this case. The equation has two real roots ( one positive and one negative) where the graph intersects the x-axis. Since a = 1 and for a > 0, the graph opens upward.

Now let's observe several graphs where d > 0.

Allowing d > 0, causes the graph to shift to the right d units. The vertex is (d, -2), so changing the value of d effects the position of the vertex, it also shifts the the right d units. However, the basic shape of the graph does not change. These equations also have two real roots.

Let's investigate what happens when d < 0.

When d < 0, the graph shifts to the left d units and the vertex (d, -2) also shifts to the left d units. The axis of symmetry is x = d. Again, the basic shape of the graph does not change.

To summarize, changing the value of d in the equation

causes the graph to shift either to the left for d < 0 or the the right for d > 0 and the vertex (d, -2) also shifts to the left or right. The graph opens upward because a =1(a>0) and the baisc shape of the graph is not effected by changing d. There are two real roots for each equation.

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