Math 122, Special Graphing Cases of Sine and Cosine

Once you have learned about the basic shapes of the trig graphs and how to graph trig functions with transformations, you are ready to handle some special graphing cases. Your textbook goes over basically two special cases:

1. The case where you are adding multiples of two trig functions.
2. The case where you are multiplying a trig function by some other well known function.

Your textbook already explains case 1 in pretty good detail on pages 435 - 437 so I will focus on case 2.

First you need to recall the shapes of some basic graphs such as: (Click here after you draw each graph to see if you got it correct. Worry only about basic shape, not accuracy or scale.)

y = x	y = x2	y = 1/x
y = bx, b > 1	y = bx, b < 1	y = x 3

Now you need to realize that since sine and cosine are always bounded between 1 and -1, then if you multiply either of these functions by a number, c that is greater than one, then the result will be a graph that is bounded between c and -c.

-1 ≤ cos x ≤ 1
mult through by 5 and get: -5 ≤ 5cosx ≤ 5

Example: Graph y = 5cos x and graph the boundary graphs.

The same sort of thing happens if you multiply a function by another function that is always positive. For example, look at the graph of y = |x|cos x.

-1 ≤ cos x ≤ 1
mult through by |x| and get: -|x| ≤ |x|cosx ≤ |x|

Notice that the graph touches the upper boundary everywhere that the cosine graph touched the upper boundary of y = 1. Also the graph touches the lower boundary everywhere that cosine graph touched the lower boundary of y = -1. Also all of the zeros are still in there original places since 0 × |any number| = 0. Name some exact points on the graph where the graph touches its boundary graph and where it crosses the x-axis and then click here.

Of course, you don't have to multiply the trig function by a positive number. We could have: y = xcosx. We just have to be careful when x is negative. The right hand side of the graph will exactly match the graph above, but the left hand side will be like the graph above except reflected through the x-axis, since multiplying through by a negative flips things around.

Name some exact points on this graph and click here.

In summary, when you are graphing these kinds of problems:

1. Make sure you draw in your boundary graphs as dotted lines, since they are not part of the answer, but only guides.
2. Keep your graph between the boundaries everywhere since the graphs values must always lie between the boundaries.
3. Make the graph touch the boundary graphs in all of the right places.
4. In many problems it is more important to get the correct shape than it is to draw to scale (sometimes it is impossible to do both), therefore be able to make these graphs by hand instead of relying on your calculator.
5. You should be able to name several exact points on the graphs.

Try making hand drawn graphs of y = f(x)sinx and y = f(x)cosx where f(x) is any of the basic graphs above or the absolute values of the basic graphs above. Check your solutions by graphing in your calculator. When does it make a difference as to whether you use one of the basic graphs above or the absolute value of a basic graph? Since we aren't concerned about scale, you may have to zoom in and/or out a lot to check your solution on your calculator.