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

Derivation of Quadratic Equation

From the general form of a quadratic equation, it is possible to derive the quadratic formula used to solve equations of this type.

1. Start with the general form of a quadratic equation 2. Subtract 3. Factor an 4. Complete the square on the left hand side by adding ( 5. Expand ( 6. Take the square root of both sides of the equation (Don't forget the +/- !!) 7. Simplify the denominator of the right hand side 8. Subtractb/2a from both sides |