Before we begin to graph trigonometric functions, you might need to review the transformations that allow you to graph functions quickly and the points that make up one cycle of the "toolkit" functions,y = sin x and y = cos x. Remember, "i" stands for an x-intercept, "M" stands for a maximum point, and "m" stands for a minimum point.
|y = sin x||y = cos x|
|i (0, 0)||M (0, 1)|
|M (p/2, 1)||i (p/2, 0)|
|i (p, 0)||m (p, -1)|
|m (3p/2, -1)||i (3p/2, 0)|
|i (2p , 0)||M (2p , 1)|
If you need to refresh your memory on the basic transformations, go to the now. You can print out a copy to use as you work.
There are two ways you can go about graphing a sine or cosine function. Let's use the example of y=-3cos(0.5x + p) - 1 and use both methods.
Use the points for the "toolkit" function shown above and do the transformations to those points to obtain 1 cycle of your graph and then repeat the cycle in either direction. For our example we would the multiply all "y" values by -3 and then subtract 1 from each product. For the "x" values, we would first subtract p from all "x" values and then multiply the result by 2. Remember, with the "y" values, you do the operation you see and use the correct order-of-operations. With the "x" values, you do the inverse operation and in the reverse order-of-operations.
|New "x"||x||y=cos x||New "y"|
|-2p = 2(0-p )||0||1||-3(1)-1 = -4|
|-p = 2(p /2- p)||p/2||0||-3(0)-1 = -1|
|0 = 2(p - p)||p||-1||-3(-1)-1 = 2|
|p = 2(2p /2- p)||3p/2||0||-3(0)-1 = -1|
|2p = 2(2p - p)||2p||1||-3(1)-1 = -4|
Use the transformations on the "toolkit" y = cos x. For our example, we know the graph has been moved p places to the left first and then stretched horizontally by a factor of 2. Then the graph has been vertically stretched by 3, the "-" in front of the 3 reflects the graph around the x-axis and then the whole graph has been moved down 1 unit on the y-axis. A few reminders that will help you sketch your graph.
- When deciding what scale to use on the x-axis take the period of the function and divide by 4. Remember the period is 2p/b, so in this example, it would be 2p /.5 = 4p. Now divide this by 4 (because there are 4 spaces between the five points that make up a cycle) to get a scale of p.
- The phase shift is -c/b, so for our example, - p/.5 or -2p .
- When the graph has been reflected around the x-axis as our example has, the maximum and minimum points of the "toolkit" are reversed.