**Lisa Brock**

** **

**Assignment 3**

**The Affects of a, b, and c on **

To best examine the
affects of a, b, and c on the curve , we
must examine each variable individually while holding the others constant.

Let's begin with *a*. Let *b* = 1 and *c* = 1. This results in the
equation . Graph this equation for different
values of *a*.

Each graph goes through
the point (0,1). When *a* = 0, the graph is a line instead of a
parabola. When a = 0, the term becomes zero, turning the equation
into a linear equation. When *a* is positive, the graph is concave up. When *a* is negative, the graph is concave down. As the absolute value of *a* increases the parabola becomes more steep and
narrow. Also, the position of the
vertex changes as *a* is varied.

Therefore, *a* determines the direction of concavity, the
steepness of the parabola, and the position of the vertex.

Now let's take a look at *b*. Let
a = 1 and c=1. This results in the equation . Graph this equation for different
values of *b*.

Each parabola goes through
(0, 1). Each graph is concave
up. Each graph is of equal
steepness. The vertex shifts as *b* changes.
As *b* increases in the
positive direction, the graph shifts farther to the left. As *b* increases in the negative direction, the graph
shifts farther to the right.

Therefore, *b* has an affect on the position on the vertex. Since the value of *a* also has an affect on the position of the vertex,
the values of both *a* and *b* will determine the position of the vertex.

Let's see what affect c
has on the graph. Let a = 1 and b
= 1. This results in the equation . Graph this equation for different
values of *c*.

Each graph is concave
up. Each graph is of equal
steepness. The vertex has shifted
vertically but not horizontally.

Therefore, *c* affects the y-coordinate of the vertex.

In summary, *a*, *b*,
and *c* all determine the
position of the vertex. *c* appears to affect only the y-coordinate of the
vertex. *b* appears to affect both coordinates of the
vertex. *a* appears to affect both coordinates of the
vertex. *a* also determines the direction of concavity and the
steepness of the parabola.

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