The first parameter we will consider is a. How does the value of a affect the function

We will evaluate the function

So when a=1, So the y-coordinate of the point is not affected by the value of a. What about if a>1? The following table shows the function with varying values of a and evaluated at .

As we can see from the table above, as the value of a increases, the y-coordinate increases by a factor of a. Now we will look at how a affects the graph of

Now we know if a>1, then the graph is stretched vertically and the range of the function is affected. What if a=-1?

We will evaluate the function

So an a value of -1 simply makes the y-coordinate negative. Now we will consider when a<-1. The following table shows the function with varying values of a and evaluated at .

Before we look at the graphs of these function, we will look specifically at the difference between the graphs of

So the only difference between and is that the y-coordinates are multiplied by a factor of -1. That is, the graph of is reflected about the x-axis.

Now we will look at the graph of with the different a values we used in the second table.

From the algebra and graphs above, we can tell that if a<-1 then the graph of is reflected about the x-axis.

What about if -1 < a < 1? The following table shows the function evaluated at

The above table shows us that if -1 < a < 1 then the graph is compressed vertically and the range of the function is affected. Now we will look at the graphs of these functions.

We have now found that if a is negative then the graph of is reflected over the x-axis. If a > 1 or a< -1 then the graph is stretched vertically by a factor of If -1 < a < 1 then the graph is compressed vertically by a factor of

Next, we will consider the effect of parameter b on the function First, let's recall that

So when b = 1,

We will investigate the effects of b by looking at the graphs of the functions with different b values.

From the graph we can tell that if b > 1 then the graph is compressed horizontally. As mentioned before, the period for the function is , meaning the function repeats every (remember ≈6.2831).

Since increasing the value of b causes the function to compress horizontally, how will this affect the period?

As we mentioned before, once the graph has one bump above the x-axis and one bump below the x-axis then the function has completed one period. To see this more clearly, below is a graph of just the function.

The green lines marks each time there is a bump below the x-axis and a bump above the x-axis. So between each green line we have a period of the function. So we have a period of π for the function .

Now let's look at the . The graph of the function is shown below.

Again, between each of the green lines, we have a bump below the x-axis and a bump below the x-axis. Therefore we have a period between the green lines and the period is

Before we draw some conclusions about the effects of parameter b we will consider when 0 < b < 1. Let's look at the graph of some functions with 0 < b < 1.

So it appears that if 0 < b < 1 then function stretches out and the period of the function is increased. repeats every so the period is 4π. repeats every so the period is 6π.

Here is a summary of what we have found so far concerning the parameter b:

As the value of b increases by a factor of 2, the period of the function decreases by the same factor and as the value of b decreases, the period of the function increases. Therefore the period of the function is equal to then the function will compress horizontally and if then the function is stretch horizontally.

is an odd function so . Therefore, we do not need to address if b < 1 because we have really already addressed this when we discussed the parameter a.

Finally, we will look at how the parameter c affects the function . We will start by looking at the function when c = -1, c = -.5, c = .5, and c = 1.

We can see the functions have the same period and amplitude. The only is that the function is shifted to the right or to the left. If c < 0 the function is shifted to the right and if c > 1 then the function is shifted to the left.

We can rewrite the function Once we have written our function in this format, the horizontal shift of the function is

So for the function , the parameter a affects the amplitude of the function, parameter b affects the period of the function, and the parameter c affects the horizontal shift of the function.

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