# 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 from**Lesson 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.