Pre-Class Problems16 for Friday, October19

These are the type of problems that you will be working on in class. These problems are fromLesson 8.

Objective of the following problems: To use a sketch of the graph of two cycles of the sine and cosine functions in order to sketch two cycles of the respective cosecant and secant functions and label the numbers on the x- and y-axes.

1.Identify the amplitude, period, and phase shift. Sketch two cycles of the graph of the following functions on the same side of the y-axis. Label the numbers on the x- and y-axes as needed. You only need to label where each cycle begins and ends.

a. b. c.

d. e.

f. g.

h. i.

Additional problems available in the textbook: Page 191 …19 - 23, 25, 26, 29, 33, 34, 35, 37, 41, 42, 45, 46, 48. Don’t use a graphing utility for Problems 41, 42, 45, 46, and 48. Examples 4 and 5 on page 188.

Solutions:

1a.

NOTE: We will first need to sketch two cycles of the graph of the function .

Amplitude of the sine function: 5

Amplitude of the cosecant function: None

Period ofboth functions: =

Phase Shift of both functions: None

x x

NOTE: The vertical asymptotes of this cosecant function occur at the x-intercepts of the sine function.

NOTE: The local maximums of the cosecant function occur at the local minimums of the sine function. Also, the local minimums of the cosecant function occur at the local maximums of the sine function.

1b.

NOTE: We will first need to sketch two cycles of the graph of the function .

Amplitude of the cosine function:

Amplitude of the secant function: None

Period of both functions: = =

Phase Shift of both functions: None

x x

NOTE: The vertical asymptotes of this secant function occur at the x-intercepts of the cosine function.

NOTE: The local maximums of the secant function occur at the local minimums of the cosine function. Also, the local minimums of the secant function occur at the local maximums of the cosine function.

1c.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: We also have that

Amplitude of the sine function :

Amplitude of the cosecant function : None

Period of both functions: = = = 12

Phase Shift of both functions: None

12 24 x

NOTE: The vertical asymptotes of this cosecant function occur at the x-intercepts of the sine function.

NOTE: The local maximums of the cosecant function occur at the local minimums of the sine function. Also, the local minimums of the cosecant function occur at the local maximums of the sine function.

1d.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: Because of the multiplication by the number , the cosine (and the secant) cycles will be inverted.

Amplitude of the cosine function : 11

Amplitude of the secant function : None

Period of both functions: = = =

Phase Shift of both functions: None

x x

NOTE: The vertical asymptotes of this secant function occur at the x-intercepts of the cosine function.

NOTE: The local maximums of the secant function occur at the local minimums of the cosine function. Also, the local minimums of the secant function occur at the local maximums of the cosine function.

1e.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: Because of the multiplication by the number , the sine (and the cosecant) cycles will be inverted.

Amplitude of the sine function : 4

Amplitude of the cosecant function : None

Period of both functions:

Phase Shift of both functions: units to the left

NOTE: The cycle will start at because of the shift.

NOTE: Starting Point + Period = = = .

Thus, the cycle will end at .

NOTE: Since the cycle starts at and ends at , the cycle will have to cross the y-axis. With the up and down oscillation of the sine cycle, I do not want to cross the y-axis. Remember, where one cycle ends, another cycle begins. So, we will sketch the cycle that starts at as our first cycle and the cycle that comes after this one as our second cycle.

NOTE: Starting Point 1 + Period = = = . Thus, our first sketched cycle will end at .

Starting Point 2 + Period = = = . Thus, the second sketched cycle will end at .

NOTE: The vertical asymptotes of this cosecant function occur at the x-intercepts of the sine function.

NOTE: The local maximums of the cosecant function occur at the local minimums of the sine function. Also, the local minimums of the cosecant function occur at the local maximums of the sine function.

1f.

NOTE: We will first need to sketch two cycles of the graph of the function .

Amplitude of the cosine function : 1

Amplitude of the secant function : None

Period of both functions: = 2

Phase Shift of both functions: units to the right

NOTE: The first cycle will start at because of the shift.

NOTE: Starting Point 1 + Period = = = . Thus, the first cycle will end at .

Starting Point 2 + Period = = = . Thus, the second cycle will end at .

x x

NOTE: The vertical asymptotes of this secant function occur at the x-intercepts of the cosine function.

NOTE: The local maximums of the secant function occur at the local minimums of the cosine function. Also, the local minimums of the secant function occur at the local maximums of the cosine function.

1g.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: The was obtained by = = .

NOTE: Because of the multiplication by the number , the sine (and the cosecant) cycles will be inverted.

Amplitude of the sine function :

Amplitude of the cosecant function : None

Period of both functions:

Phase Shift of both functions: units to the right

NOTE: The first cycle will start at because of the shift.

NOTE: Starting Point 1 + Period = = = . Thus, the first cycle will end at .

Starting Point 2 + Period = = = . Thus, the second cycle will end at .

NOTE: The vertical asymptotes of this cosecant function occur at the x-intercepts of the sine function.

NOTE: The local maximums of the cosecant function occur at the local minimums of the sine function. Also, the local minimums of the cosecant function occur at the local maximums of the sine function.

1h.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: The was obtained by = = = .

NOTE: Because of the multiplication by the number , the cosine (and the secant) cycles will be inverted.

Amplitude of the cosine function :

Amplitude of the secant function : None

Period of both functions: = =

Phase Shift of both functions: units to the left

NOTE: The cycle will start at because of the shift.

NOTE: Starting Point + Period = = =

. Thus, the cycle will end at .

NOTE: Since the cycle starts at and ends at , the cycle will have to cross the y-axis. With the up and down oscillation of the cosine cycle, I do not want to cross the y-axis. Remember, where one cycle ends, another cycle begins. So, we will sketch the cycle that starts at as our first cycle and the cycle that comes after this one as our second cycle.

NOTE: Starting Point 1 + Period = = = . Thus, our first sketched cycle will end at .

Starting Point 2 + Period = = = . Thus, the second sketched cycle will end at .

| | | |

x x

1i.

NOTE: We will first need to sketch two cycles of the graph of the function .

NOTE: The was obtained by = = = .

Amplitude of the sine function :

Amplitude of the cosecant function : None

Period of both functions: = = =

Phase Shift of both functions: units to the right

NOTE: The first cycle will start at because of the shift.

NOTE: Starting Point 1 + Period = = = . Thus, the first cycle will end at .

Starting Point 2 + Period = = = . Thus, the second cycle will end at .

x x

Reducing the fraction, we obtain our required sketch.

x x

Solution to Problems on Pre-Exam 2:

6.Sketch the graph of two cycles of the following function on the same side of the y-axis. Label the numbers on the x- and y-axes as needed. Label where the cycles begin and end. (6 pts.)

Amplitude: None

Period = 12

Period = = = = 12Phase Shift: None

NOTE: Because of the multiplication by the number , the cosine (and the secant) cycles will be inverted.

12 24 x