## Investigating the graphs of y=asin(bx+c)

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In this this assignment, we are investigating the graphs of y=asin(bx+c) for different values of a, b, and c. The first investigation that I would like to perform is to see what happens to the graph of y=asinx(bx+c) when we let a = b = 1 and change c. Hence, we will be investigating the graphs of the following equation:

y=sin(x+c), where c varies from -1 to 1.

The following graphs show what happens when we change c.

By observing these graphs, one can see that if we keep a=b=1 and change c the normal sine curve is shifted left or right depending on if c is positive or negative.
Now lets see what happens when we change a and let b=1 and c=0. Now we are graphing the equation

y=asin(x) for a varying from -1 to 1.

From these graphs, one comes to the conclusion that any change in a will change the amplitude of the sine curve.
So far we have found out what happens to the graph of y=asin(bx+c) when we change either a or c and keep the other two constant. Now we need to see what happens when we change b and let a=1 and c=0. Now we are graphing the following equation:

y=sin(bx) where b varies from -3/2 to 3/2.

From these graphs, one can see that b stretches or shrinks the period of the sine curve. If -1<b<1, then the period of the sine curve is larger. If b>1 or b<-1, then the period is smaller.
What happens to the graph when we change two of the variables and keep the third one constant? Will let's find out. First, let's examine what happens when we change a and b and let c=0. The equation that we are using now is

y=asin(bx).

From looking at these graphs, we can see that the graph has been changed in the same manner as the above graphs. The only difference is that both the period and the amplitude have changed simultanously.
Now, let b=1 and change a and c. So we are investigating the equation

y=asin(x+c).

As you can see, when we change a and c the graph shifts and the amplitude changes.
What changes would you expect if we changed all of the variables? Let's find out. Now we are experimenting with the equation

y=asin(bx+c).

As one can see from the above graph, changing a, b, and c simultanously changes the graph in three different ways. The graph is shifted left or right and the amplitude and period changes according to the changes of a and b.
In the equation y=asin(bx+c), the variables a, b, and c all change the location and size of the original sin graph. "a" changes the amplitude,"b" changes the period, and "c" shifts the graph left or right.