In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don't know its exact value at a given instant. For example, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values. This project describes a simple example of a function with a maximum value that depends on two-equation coefficients. We will see that maximum values can depend on several factors other than the independent variable x.

1. Consider the graph of the function y = sin x + cos x. Describe its overall shape.

• Is it periodic?
• How do you know?

2. Using a graphing calculator or other graphing device, estimate the x- and y-values of the maximum point for the graph (the first such point where x > 0). It may be helpful to express the x-value as a multiple of .

3. Now consider other graphs of the form y = A sin x + B cos x for various values of A and B.

• Sketch the graph when A = 2 and B = 1, and, find the x - and y-values for the maximum point. (Remember to express the x-value as a multiple of , if possible.)
• Has it moved?

4. Repeat and sketch the graph for A = 1, B = 2.

• Is there any relationship to what you found in part (2)?

5. Explain what you have discovered from completing this activity using details and examples.