Modeling with Periodic Functions

A student is given that point P(a, b) lies on the terminal ray of angle , which is between radians and 2 radians. The student uses the steps below to find cos . Step 1Find the quadrant in which P(a, b) lies:P(a, b) is in Quadrant IV.Step 2Use the point and the Pythagorean theorem to determine the value of r:, but since r must be positive, .Step 3Determine cos ., where a and b are positive.Which of the following explains whether the student is correct?

A

The student made an error in step 3 because a is positive in Quadrant IV; therefore, .

B

The student made an error in step 3 because .

C

The student made an error in step 2 because r is negative in Quadrant IV; therefore, .

D

The student made an error in step 2 because using the Pythagorean theorem gives .

2

Free Preview

A circle has a central angle measuring radians that intersects an arc of length 18 cm. What is the length of the radius of the circle? Round your answer to the nearest tenth. Use 3.14 for .

A

3.7 cm

B

4.9 cm

C

14.3 cm

D

15.4 cm

3

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 a.m. and 3:30 p.m., with a depth of 3.25 meters, while high tides occur at 7:45 a.m. and 11:15 p.m., with a depth of 8.75 meters. Which of the following equations models d, the depth of the water in meters, as a function of time, t, in hours? Let t = 0 be 12:00 a.m.

A

d = negative 3.75 cosine (StartFraction 4 pi Over 31 EndFraction t) + 5

B

d = negative 3.75 cosine (StartFraction 4 pi Over 29 EndFraction t) + 5

C

d = negative 2.75 cosine (StartFraction 4 pi Over 31 EndFraction t) + 6

D

d = negative 2.75 cosine (StartFraction 4 pi Over 29 EndFraction t) + 6

4

A student uses the equation to represent the speed, s, in feet per second, of a toy car driving around a circular track having an angle of incline , where . To solve the problem, the student used the given value of to find the value of and then substituted the value of in the equation above to solve for s. What is the approximate value of s, the speed of the car in feet per second?

A

5.3

B

7.5

C

9.2

D

28.3

5

Which of the following are in correct order from greatest to least?

A

, 330°, , ,

B

, , , , 330°

C

, , , , 330°

D

330°, , , ,

6

Which statement accurately describes how adding a number, n, to the function affects its graph?

A

There is a vertical shift of n units.

B

The x-intercepts will shift n units.

C

There is a change in amplitude of n units.

D

The range will change by a factor of 2n units.

7

Which transformations are needed to change the parent sine function to the sine function below?

A

vertical compression of , horizontal stretch to a period of , vertical shift of 1 unit up, phase shift of units left

B

vertical stretch of 2, horizontal compression to a period of , vertical shift of 2 units up, phase shift of units left

C

vertical stretch of , horizontal compression to a period of , vertical shift of 1 unit up, phase shift of units left

D

vertical compression of , horizontal stretch to a period of , vertical shift of 1 unit down, phase shift of units left

8

What is the approximate length of arc s on the circle below? Use 3.14 for . Round your answer to the nearest tenth.

A

5.8 ft

B

6.3 ft

C

27.5 ft

D

69.1 ft

9

Which transformations are needed to change the parent sine function to the sine function below?

A

vertical compression of , horizontal stretch to a period of , vertical shift of 1 unit up, phase shift of units left

B

vertical stretch of 2, horizontal compression to a period of , vertical shift of 2 units up, phase shift of units left

C

vertical stretch of , horizontal compression to a period of , vertical shift of 1 unit up, phase shift of units left

D

vertical compression of , horizontal stretch to a period of , vertical shift of 1 unit down, phase shift of units left

10

Which of the following is true for f(x) = –2sin(x) – 3?

A

The range of the function is the set of real numbers .

B

The graph of the function is the graph of f(x) = –2sin(x) shifted 3 units up.

C

The amplitude of the function is 2.

D

The period of the function is .

11

Which values for have the same reference angles?

A

, ,

B

, ,

C

, ,

D

, ,

12

Which formula gives the zeros of y = sin(x)?

A

for any positive integer k

B

for any integer k

C

for any positive integer k

D

for any integer k

13

A student uses the equation to represent the speed, s, in feet per second, of a toy car driving around a circular track having an angle of incline , where . To solve the problem, the student used the given value of to find the value of and then substituted the value of in the equation above to solve for s. What is the approximate value of s, the speed of the car in feet per second?

A

5.3

B

7.5

C

9.2

D

28.3

14

What is the approximate length of arc s on the circle below? Use 3.14 for . Round your answer to the nearest tenth.

A

5.8 ft

B

6.3 ft

C

27.5 ft

D

69.1 ft

15

Which of the following are in correct order from greatest to least?

A

, 330°, , ,

B

, , , , 330°

C

, , , , 330°

D

330°, , , ,

16

Which function is the same as ?

A

y = 3 sine (2 (x + StartFraction pi Over 4 EndFraction)) minus 2

B

y = negative 3 sine (2 (x + StartFraction pi Over 4 EndFraction)) minus 2

C

y = 3 cosine (2 (x + StartFraction pi Over 4 EndFraction)) minus 2

D

y = negative 3 cosine (2 (x + StartFraction pi Over 2 EndFraction)) minus 2

17

A student is given that point P(a, b) lies on the terminal ray of angle , which is between radians and 2 radians. The student uses the steps below to find cos . Step 1Find the quadrant in which P(a, b) lies:P(a, b) is in Quadrant IV.Step 2Use the point and the Pythagorean theorem to determine the value of r:, but since r must be positive, .Step 3Determine cos ., where a and b are positive.Which of the following explains whether the student is correct?

A

The student made an error in step 3 because a is positive in Quadrant IV; therefore, .

B

The student made an error in step 3 because .

C

The student made an error in step 2 because r is negative in Quadrant IV; therefore, .

D

The student made an error in step 2 because using the Pythagorean theorem gives .

18

The average daily temperature, t, in degrees Fahrenheit for a city as a function of the month of the year, m, can be modeled by the equation , where m = 0 represents January 1, m = 1 represents February 1, m = 2 represents March 1, and so on. Which equation also models this situation?