# Parabola

by

## B.J Kim

Explore y=ax^2+bx+c

I fixed b=1, c=2 , so we can get y=ax^2+x+2

## What happens to the parabola as a varies?

##

## Observation

When a=0, the parabola changed into a line, which is y=x+2

If a>0, the parabola opens up. The vertex of the parabola has the minimum value.

If a<0, the parabola opens down. The vertex of the parabola has the maximum value.

As |a| increases, the parabola becomes narrow. On the contrary, As |a| decreases, the parabola becomes wider.

**What happens to the parabola as b varies?**

I fixed a=1 and c=1. so, we can have, y=x^2+bx+1

As b varies from -5 to 5, note that the vertex of the parabola seems to trace along a parabola that opens down.

The vertex traces along the reflection of the original parabola when b=0.

When a and b are the same sign, the symmetric axis lies on the left side of y-axis.

When a and b are different sign, the symmetric axis exists in the right side of y axis.

The reason why follows.

## What happens to the parabola as c varies?

c is the value of y when x=0. We say this y intercept.

I consider y=ax^2+bx+c, where a=1 or a=-1 and b=0

c varies from -5 to 5

As c increase, the parabola shifts up along the y axis.

As c decrease, the parabola moves down vertically.