To examine the graph of y = cos x, I will examine y = A cos (Bx +C) for different values of A, B, and C. This will allow me to make a generalization for the values of A, B, and C and thus will know how to graph a function of y = cos x quickly.
Let's us first look at the graph y = cos x. This is were A and B equal 1 and C equals 0. This is the graph that we will compare other graphs to.
We can see from the graph above what the differences are when we change the value of "A" in y =A cosx. Notice that the magnitude of the curves is what is affected. This is called the amplitude of the curve. So A affects amplitude. To figure out the amplitude of a curve we can use this easy formula.
Now that we know what the magnitude of the amplitude is, we need to decide if the sign of A is of any importance. Notice that the last three graphs are negative values for A. Does this make a difference? We can see that they reflect the graphs with the same numerical values (only positive) about the x-axis.
It appears that B affects the period of the curve. To see if this is true, lets graph some curves where the value of B is less than zero.
We can see that in fact, B does affect the period of the curve. It takes 1/B times to complete a period of a curve. If B is equal to 1, then it takes 2pi to complete a period. If B is equal to 2, then it takes only pi to complete a period. If B is equal to 1/2, then it takes only 4pi to complete a period, twice as long as a normal period. Once again we also see that a negative only reflects the curve about the x-axis.
Automatically we can see that the actual picture of the graph does not change, it only shifts. If C is positive it shifts to the right, if C is negative it shifts to the left. Thus C affects the horizontal displacement (or shift) of the graph.