**1. Construct graphs for the parabola**

**for different values of a, b, and c.**

**Let's look at the equation**

**and let a have the
following values: {-10, -5, 0, 5, -10}**

**so here are our new equations:**

**and here are our graphs:**

**First of all, notice that if a is
positive, then the graph opens up and if a is negative
then the graph opens down. Also, as | a | gets larger,
the graph gets skinnier or looks like it is stretched.**

**Let's look at the equation**

**and let b have the following values: {-4, -2, 0, 2, 4}**

**so here are our new equations:**

**and here are our graphs**

**Notice first that the graphs
are reflected around the y-axis. For example, the purple graph
is reflected in the light blue graph and the red graph is reflected
in the green graph. However, note the interesting fact that if
b is positive then the vertex has a negative x value
and if b is negative then the vertex has a positive x
value. Lastly, notice that all graphs go through the point (0,
2). This is because when the x-value equals 0, then the y-value
equals c, which in this case is 2.**

**Let's look at the equation**

**and let c have the following values: {-4, -2, 0, 2, 4}**

**so here are our new equations:**

**and here are our graphs**

**Notice first that each graph
has the same x-value for its vertex: -3/4. The reason for this
is because each equation is the same except for the constant and
the constant plays no part in determining where the minimum or
maximum is in a graph. Minimums and maximums are determined by
finding the first derivatives of an equation. In this case, each
graph has the same derivative: 4x + 3 = 0 which yields x = -3/4.
So, if we start with our blue graph where c = 0, we see
that each graph shifts up or down depending on its c value.
For instance, the light blue graph is 4 units above the blue graph
because c = 4.**