Problems Which Are Due at the Beginning of Class

Pre-Class Problems10 for Monday, March11

Problems which are due at the beginning of class:

1.Identify the amplitude, period, and phase shift. Sketch one cycle of the graph of the following functions. Label the numbers on the x- and y-axes as needed.

a. b.

2.One website that you used for these pre-class problems other than mine.

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

You can go to the solution for each problem by clicking on the problem letter.

Objective of the following problems: To sketch the graph of one cycle of the sine and cosine functions with a phase shift and label the numbers on the x- and y-axes.

1.Identify the amplitude, period, and phase shift. Sketch one cycle of the graph of the following functions. Label the numbers on the x- and y-axes as needed.

a. b.

c. d.

e. f.

g. h.

Additional problems available in the textbook: Only sketch one cycle for the following. Page 181 … 49, 50, 51, 52, 59, 60, 67, 68.

Don’t use a graphing utility for Problems 67 and 68. Examples 4 and 5 on page 177.

Solutions:

1a.

Amplitude = 2Period =

Phase Shift: units to the right

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

NOTE: Starting Point + Period = = = .

Thus, the cycle will end at .

period = = =

y

2

x

x

NOTE: Starting Point + period = = = .

Reducing the fractions, we obtain our required sketch.

y

2

x

x

Back to Problem 1.

1b.

Amplitude = Period =

Phase Shift: 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, I will sketch the cycle that starts at .

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

y

x

x

period = =

NOTE: Starting Point + period = = .

Reducing the fractions, we obtain our required sketch.

y

x

x

Back to Problem 1.

1c.

NOTE: Because of the multiplication by the number , the cycle will be inverted.

Amplitude = Period =

Phase Shift: 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, I will sketch the cycle that starts at .

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

y

|x

x

period = = =

NOTE: Starting Point + period = = = .

Reducing the fractions, we obtain our required sketch.

y

|x

x

Back to Problem 1.

1d.

NOTE: The was obtained by = = .

NOTE: Because of the multiplication by the number , the cycle will be inverted.

Amplitude = 4Period = = =

Phase Shift: units to the right

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

NOTE: Starting Point + Period = = = .

Thus, the cycle will end at .

y

4

x

x

period = = =

NOTE: Starting Point + period = = = .

Back to Problem 1.

1e.

NOTE: The was obtained by = = .

NOTE: Because of the multiplication by the number , the cycle will be inverted.

Amplitude = Period = =

Phase Shift: 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, I will sketch the cycle that starts at .

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

y

| x

period = = =

NOTE: Starting Point + period = = = .

Reducing the fraction , we obtain our required sketch.

y

| x

Back to Problem 1.

1f.

NOTE: The was obtained by = = =

.

Amplitude = Period = = =

Phase Shift: units to the right

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

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

y

x

x

period = = = =

NOTE: Starting Point + period = = =

.

NOTE: You could use the prime factorization of 15 and 18 in order to find the least common denominator of these two denominators.

LCD(15, 18) = = 90

Reducing the fractions, we obtain our required sketch.

y

x

x

Back to Problem 1.

1g.

Amplitude = 6Period =

Phase Shift: units to the right

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

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

y

6

x

x

period = = =

NOTE: Starting Point + period = = = .

Reducing the fraction , we obtain our required sketch.

y

6

x

x Back to Problem 1.

1h.

NOTE: The was obtained by = = = .

Amplitude = Period = = = = 20

Phase Shift: units to the right

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

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

y

x

x

period = = 5 =

NOTE: Starting Point + period = = = .

Back to Problem 1.