5. Examine graphs of y= a sin (bx+c) for different values of a,b, and c.

I used the following equations and different values for a,b,and c to obtain my graph and conclusions.

My investigation began by taking a look at the following three sin functions.

For the above three equations, I held

bandcconstant and changed different values forawhich is the coefficient of the sin function. For y=sinx, the function appears as a normal sine function that is periodic at 2pi. The 2 and the 5 affect the graph by stretching it out taller. The 2 doubles the heighth of the y=sinx function. The 5 stretches the graph out even taller. However, all three graphs still cross the x-axis at the same points and have the same periodicity. Here is the graph of the three sin functions:

Next, I tried some more sine functions. The graphs of the functions are shown below.

For these graphs,

bwas held at 2,cat 0 and I changed the values ofa. The effect thatbhad at 2 compared to y=sin(x) was that it cut its periodic length in half. Y=sin(2x) has period of pi instead of 2pi like sin(x). Changingato have the different values had the same effect as discussed in the first three equations. It just stretched out the y=sin(2x) graph but all three graphs still crossed the x-axis at the same point.

My investigation continued on with the graphs below:

Comparing these graphs at first glance to the graph of y=sin(2x), I notice that the second equation moves the sin function to the left 1/2 space. The graph of y= 2sin(2x+1) just doubles the heighth size of the y= sin(2x+1). The 2 in front of the sin function just has the effect of stretching the graph out taller. The graphs of these functions is given below:

Next, I looked at the functions given below:

I decided to add the negative aspect to the graphs to see what effect it had on the graphs of the sine functions. For y=-sinx, the graph of this is y=sin x flipped or the inverse of y=sinx. For y=-sin(2x+1), this graph looked like the y=sin(2x+1) flipped over. Take a look at the graphs below for yourself.

Finally, I looked at the functions below to conclude my investigation:

I first compared these types of graphs to one like y=sin5x. The 5 has the effect of making the graph more compact with its periodicty coming more often than the graph of y=sinx. Next, for the first graph of study, the 2 had the effect of moving the graph up along the y-axis two spaces. For the second graph, the 10 had the effect of making the sin function even more compact than the y=sin5x graph. In addition, the + 2 had the same effect as the 2 did in the second graph. It caused the graph to be moved up along the y-axis two spaces. See the graphs below for reference.

This concludes my examination of the graphs y = a sin (bx+c) for different values of a,b,and c.