```Assignment #2
Nicole Mosteller
EMAT 6680
```

To make the new graph for part one, I experimented with different values of a, b, and c, in the basic equation for a parabola, and then I decided that the easiest way to find the new graph would be to use a bit of algebra and calculus to first find the equation of the new parabola. To find the equation of a parabola in the form

three points on the new graph are needed. Below is the explanation of how I found these three points as well as how I used these points to find arrive at the equation of the new graph.

Figure 1.

Before finding the three points for the new graph, we need to manipulate the equation for the old graph that will make our work a bit easier. First simplify the equation

to make our working equation to be

.

Point #1: The Vertex.

To find the vertex of the new graph, we can find the vertex of the new graph. From the work seen on Jadonna Brewton's Assignment #2 Write Up, we know that the vertex of a parabola can be found using the formula:

```If , then the vertex is located at the point
.
Using this formula, the vertex for the original parabola which will be shared with the new parabola becomes
.
Substituting this point into the basic form for a parabola, we get our first equation
.```

Point #2: The Y-Intercept.

The y-intercept of the original graph is at the point (0,16). To find the next point on the new parabola, we know that the y distance from the vertex of the old parabola to its y-intercept will be equal to the y distance from the vertex to the y-intercept of the new parabola. This equal distance is 21.125. Now, to find the y-coordinate of the y-intercept of the new parabola, let's take this distance, 21.125 and add it to the y-distance of the vertex from the x-axis, 5.125.

```The coordinate of the y-intercept for the new parabola is
(0,-26.25).
Substituting this point into the basic form for a parabola, we get our second equation
-26.25 = c.```

Point #3: The Point Symmetric to the Y-Intercept.

Because a parabola has a vertical line of symmetry through the vertex, we can say that each point on one side of a parabola has a "mate" on the opposite side of the line symmetry. In the original parabola, the y-intercept's (0, 16), symmetric "mate" is the point (7.5, 16). The symmetric mate for the y-intercept (0, -26.25), of the new parabola uses the same x coordinate.

```The third point for the new parabola is
(-7.5, -26.25).
This point produces the third equation
.```

We now have the materials to build the equation of the new parabola. By solving this system of equations,

```
-26.25 = c
```
```we find the solutions to be a = -2, b = 13, and c-26.25.
The equation for the new parabola is
.