Assignment 2:

Exploring

by

Margo Gonterman

The general form of a quadratic equation is:

The vertex form of a quadratic equation is:

The factored form of a quadratic equation is:

This investigation will focus on how the parameters a, b, and c in general form change the shape and orientation of the parabola.

What happens to the parabola as a varies?

• When a>0, the parabola opens up. When the parabola opens up, the vertex of the parabola is the minimum.
• When a<0, the parabola opens down. When the parabola opens down, the vertex of the parabola is the maximum.
• When a=0, the parabola degenerates into a line.
• As |a| increases, the parabola becomes more narrow.
• As |a| decreases, the parabola becomes wider.

See the animation below to watch what happens as a varies from -3 to 3.

What happens to the parabola as b varies?

As b varies from -3 to 3, note that the vertex of the parabola seems to trace along a parabola that opens down.

In fact, the vertex traces along the reflection of the original parabola when b=0.

The reflection of the parabola given by is .

Below is an animation of a parabola in standard form as b varies from -5 to 5 overlaid with its reflection when b=0.

What happens to the parabola as c varies?

• As c increases, the parabola shifts up vertically.
• As c decreases, the parabola shifts down vertically