Exercises

1. 1. What are the domains and ranges of the function and ? Do the amplitudes of and have anything to do with their domains or ranges? If so, explain.
   2. Which points on the circle in Figure 1.65 correspond to where the graph of the function crosses the x-axis? Which points on the circle correspond to where the graph the function has maximum values and minimum values?
   3. Which points on the circle in Figure 1.65 correspond to where the graph of the function crosses the x-axis? Which points on the circle correspond to where the graph the function has maximum values and minimum values?
2. In our last discussion we considered the effect of changing both the dependent and independent variables by addition and multiplication. Let us try this again for . We will work with the particular function .
   1. What is the period of f? the amplitude of f?
   2. From the algebraic form of f, describe how to modify the graph of to obtain the graph of f. How do the modifications reflect algebra? Plot the graph of f with the graph of .
   3. How many steps did you use in (b)? If you do them in a different order, do you get the same result? Explain.
3. Now let . The function g isn't written in the form of the function f of the previous problem.
   1. What is the period of g? the amplitude of g?
   2. From the algebraic form of g, describe how to modify the graph of to obtain the graph of g. How do the modifications reflect algebra? (Hint: Is this question really different from 2(b)?) Plot the graph of g with the graph of .
4. There are a number of different trigonmetric identities that you should be familiar with. One is the identity

   

   Using this identity and what we have done so far, describe the graph of in terms of the graph of . In particular, what are its period, amplitude and intercepts?