Mathematics 136 – Advanced Placement Calculus Discussion 1 September 4, 2009

A) The graph y = f(x) = 4 − x^{2} for x in the interval [−1, 2] is given in Figure 1 on the back, together with several transformed graphs. Label the graphs with the letters A, B, C, D, E in some way. Then match each equation with its graph by letter and give reasons for your choices. (Note that one of the graphs is y = f(x) and you will need to identify that one as well.)

1. y = f(2x) − 4

2. y = 2f(x + 6)

3. y = −f(x + 4)

4. y = \frac{1}{2}f(x) − 1

B) Say y = f(x) = cos(x) gives the usual graph of the cosine function for x in the domain [0, 4π]. For each of the following, write equations for the graph that is described and sketch.

1. The graph y = f(x) shifted three units down.

2. The graph y = f(x) reflected across the x axis, then shifted 2 units up.

3. The graph y = f(x) shifted \frac{π}{2} units to the left.

4. The graph y = f(x) stretched vertically by a factor of 4, then shifted up one unit.

5. The graph y = f(x) compressed, or shrunk horizontally by a factor of 2.

C) Imagine that each of the scatter plots 1, 2, 3 in Figure 2 were obtained as data from some experiment.

a. Decide what type of function you might choose as a model for the data. That is, what type

of formula would yield a graph with something like this shape? Explain your choice. b. From the information given in the plot, determine a possible formula of a function of the type

you decided on for part 1. that would fit the data reasonably well. c. Plot your function.

Assignment Group writeups due in class on Monday, September 7.

Figure 1: Plots for question A

(a) Plot 1  (b) Plot 2  (c) Plot 3

Figure 2: Scatter plots for question C