Exploring Quadratic Functions

by LaShonda Davis

In exploring quadratic functions, I wanted to graph

y = ( x -d)^2 - 2

and see want happens to the graph as d varies.

First I looked at the graph of y = x^2 -2, where d = 0.

As you see, the graph is a parabola with the vertex at (0, -2).

Let's look at the general case of the vertex form of a quadratic equation.

y = a(x-d)^2 + k

(d, k) is the vertex of the parabola. For our purposes a = 1 and k = - 2.

y = (x-d)^2 -2

So as d varies, the parabola shifts along the x-axis.

So let's look at several graphs of y= (x-d)^2 -2 on the same axes using different values for d, paying close attention to the vertices of each.

It seems that the rule holds that (d, -2) is always the vrtex. The shape does not change, only the position on the x-axis.

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