Power Function:
Exercise: Recognize and analyze the graph of the power function as the exponent "n" and the coefficient "a" vary

1. Consider a positive exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?

2. Consider a negative exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?

Problem 1

Part A

If n is even, then we have a "parabolic" shape going on. I put that in quotes because if n = 2, then we have a true parabola; however, if n is even and larger than 2, then we have a curve that looks like a parabola, but it's not exactly a perfect parabola. All of these curves have in common that the end behavior is the same for the left and right sides.

Consider the example where a = 1 and n = 2. This means

y = a*x^n

turns into

y = 1*x^2

which is the same as

y = x^2

This produces the parent parabola curve. This curve opens upward.

I recommend using Desmos to graph y = a*x^n, where the 'a' and n are parameters set up. Check out the screenshot below.

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Part B

If n is odd, then we'll get something that looks like a cubic curve. If n = 3, then we get a cubic curve. If n is odd and n > 3, then we get something that looks like a cubic curve, but it's not exactly this perfect cubic.

If we let a = 2 and n = 3, then y = a*x^n becomes y = 2x^3. The graph of which is below.

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Part C

If a > 0 then the curve is like the previous examples shown earlier.

If a < 0, then we will reflect those given earlier curves over the horizontal x axis. An example is shown below using y = -x^2. This time we have a = -1 and n = 2. In other words, I reflected the example in part A over the x axis.

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Part D

As 'a' increases, then the graph gets more narrow. It also gets vertically stretched. Consider the example of going from y = 1x^2 to y = 3x^2 as shown below.

As 'a' decreases, then the process is done in reverse: the curve gets more flatter and wider horizontally. We consider the curve to be vertically compressed or squished. So we'd go from y = 3x^2 to y = 1x^2 as an example.

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Problem 2

Part A

For all of problem 2, we'll only consider negative values of n. We want n to be even, so let's say n = -2. Furthermore, let's say a = 2.

This means y = ax^n becomes y = 2x^(-2) which is the same as y = 2/(x^2)

The curve we get is really two separate pieces. There's a gap or disconnect at x = 0 because of a division by zero error. One piece of the curve is always increasing, while the other piece is always decreasing.

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Part B

Since we want n to be odd and negative, let's say n = -1. We'll keep a = 2 the same as before.

That means y = ax^n becomes y = 2x^(-1) as shown in the diagram below. What we have is a hyperbola. This is a disconnected set of two curves that collectively entirely define this function in a visual sense. Note the gap when x = 0 due to the division by zero error.

Recall that y = 2x^(-1) is the same as y = 2/x

Again, I'm using Desmos to generate each of these graphs.

Similar to part A, the two pieces are disconnected. This curve is known as hyperbola. It's always increasing if a < 0, and it's always decreasing when a > 0.

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Part C

If 'a' is positive, then the graph of y = 2/(x^2) is entirely above the x axis (aka everything is positive). On the flip side, if 'a' is negative, then everything swaps to being negative. That will reflect the curve over the x axis to get what you see in the example below. We have a = -2 and n = 2.

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Part D

As 'a' increases, this means the graph spreads out. If 'a' decreases, then it gets more compressed.  

An example is shown below going from y = 1/x to y = 3/x which shows expansion going on (reverse it to get a compression).

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Once again, the screenshots are shown below as examples.

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