**Assignment
2:**

**Analysis
of y = ax**^{2}** + bx + c**

Presented
by:

Amber Krug

y = ax^{2}
+ bx + c

Using the above equation, the problem asks to fix two of
the values for a, b, and c and construct 5 graphs on the same axes as the third
value varies.

LetŐs start
by keeping a and b constant and vary c:

Below is the
graph of

y = x^{2} + x + c
where c = -8, -4, 0, 4, 8

From the graph, we see that the x-value of the vertex
remains constant; however, the y-value of the vertex corresponds to the
c-value.

Notice the
shape and direction of each graph are the same.

Now, letŐs
keep a and c constant and alter b:

Below is the
graph of

y = x^{2} + bx + 1
where b = -6, -3, 0, 3, 6

From the
graph, we see that the x-value and y-value of the vertex change, but the shape
and direction of each parabola is the same.

Lastly,
letŐs keep b and c constant and alter a:

Below is the
graph of

y = ax^{2} + x + 1
where a = -2, -1, 0, 1, 2

a controls
the direction of the graph. When a is negative, the graph opens to the
bottom, and when a is positive, the graph opens upward. The shape of the
graph changes with a as well.