Assignment 2: Playing With Parabolas

By Krista Floer

The problem given is graph the parabola .

i. Overlay a new graph replacing each x by (x - 4).

ii. Change the equation to move the vertex of the graph into the second quadrant.

iii. Change the equation to produce a graph concave down that shares the same vertex.

So let us start by graphing the original equation:

i. Overlay a new graph replacing each x by (x - 4):

We see that by replacing x with (x - 4), the parabola has moved to the right 4. I expected this because that is the definition of a translation in a function.

ii. Change the equation to move the vertex of the graph into the second quadrant.

I changed it correctly on my first try. Basic knowledge of translations with functions helped me. I knew that if (x - 4) moved the parabola to the right, then (x + 4) would move it to the left. I also needed to move it up. Since the original equation had - 4 and that moved the vertex down 4 spaces, then 4 would move the vertex up 4 spaces from the origin.

iii. Change the equation to produce a graph concave down that shares the same vertex.

In order to do this, we need to find the vertex of the original graph. To do this we need to find h, which is the x-coordinate, and k, which is the y-coordinate. h has an equation: . To find k, we will substitute h for x in our equation and solve. Then, take the values found for h and k, and put the equation in vertex form. The formula for the vertex form is . Once we negate this equation, it should be upside down.

We have , a = 2 and b = 3.

Thus, or -3/4 . Substituting this into our equation, we get . After simplifying this, we have . Substituting the value we found for y in for k and the value we found for h into our vertex form equation, we get . This graph should coincide with the original equation. Let's check:

Our equations yield the same graph. Now we can proceed to try to flip the blue equation upside down. I know that in order to reflect a graph over a line parallel to the x-axis, a negative is involved. In Assignment 1 we saw that a negative in front of flipped it over. I used this to inform me about where to put my negative.

Thus, we can see that we have indeed produced a graph that is concave down that shares the same vertex.