Worksheet: Periodic Functions

In this worksheet, we will practice deriving the amplitude, period, midline, and horizontal and vertical shifts of a trigonometric function from its equation.

Q1:

Which function is represented on the graph?

  • A 𝑦 = 2 π‘₯ s i n
  • B 𝑦 = π‘₯ s i n
  • C 𝑦 = π‘₯ c o s

Q2:

What is the maximum value of the function 𝑓 ( π‘₯ ) = π‘Ž ( 𝑏 ( π‘₯ βˆ’ β„Ž ) ) + π‘˜ s i n ?

  • A π‘Ž
  • B 2 πœ‹ 𝑏
  • C 𝑏
  • D π‘Ž + π‘˜
  • E β„Ž

Q3:

What is the amplitude of the function 𝑓 ( π‘₯ ) = π‘Ž ( 𝑏 ( π‘₯ βˆ’ β„Ž ) ) + π‘˜ s i n ?

  • A π‘Ž + π‘˜
  • B 2 πœ‹ 𝑏
  • C 𝑏
  • D π‘Ž
  • E β„Ž

Q4:

Simplify c o s ( 1 8 0 βˆ’ πœƒ ) ∘ .

  • A s i n πœƒ
  • B c o s πœƒ
  • C βˆ’ πœƒ s i n
  • D βˆ’ πœƒ c o s

Q5:

What is the midline of 𝑦 = π‘₯ s i n ?

  • A 𝑦 = 5 3 4
  • B 𝑦 = 5 4
  • C 𝑦 = πœ‹ 7
  • D 𝑦 = 0
  • E 𝑦 = πœ‹ 7 + 1 2

Q6:

The figure shows the graph of 𝑦 = π‘₯ s i n .

Which of the following is a graph of 𝑦 = 2 π‘₯ s i n ?

  • A
  • B
  • C
  • D
  • E

Q7:

The figure shows the graph of 𝑦 = π‘₯ s i n .

Which of the following is a graph of 𝑦 = ( π‘₯ ) + 2 s i n ?

  • A
  • B
  • C
  • D
  • E

Q8:

Which of the plots shown in the graph represents the tangent function?

  • A the blue plot
  • B the green plot
  • C the red plot

Q9:

What is the amplitude of the function 𝑓 ( π‘₯ ) = π‘Ž ( 𝑏 π‘₯ βˆ’ 𝑐 ) + π‘˜ c o s ?

  • A 𝑐
  • B 𝑏
  • C π‘Ž + π‘˜
  • D π‘Ž
  • E 2 πœ‹ 𝑏

Q10:

Let 𝑓 ( π‘₯ ) = 2 π‘₯ s i n . What is the smallest positive value of 𝑃 for which 𝑓 ( π‘₯ + 𝑃 ) = 𝑓 ( π‘₯ ) holds?

  • A 2 πœ‹
  • B πœ‹ 2
  • C πœ‹ 3
  • D πœ‹
  • E 3 πœ‹ 2

Q11:

Simplify s i n ( 1 8 0 βˆ’ πœƒ ) ∘ .

  • A c o s πœƒ
  • B βˆ’ πœƒ s i n
  • C βˆ’ πœƒ c o s
  • D s i n πœƒ

Q12:

Simplify s i n ( 3 6 0 βˆ’ πœƒ ) ∘ .

  • A c o s πœƒ
  • B s i n πœƒ
  • C βˆ’ πœƒ c o s
  • D βˆ’ πœƒ s i n

Q13:

The figure shows the graph of a periodic function.

The height of the function’s midline was doubled. Which of the following is the graph of the resulting function?

  • A
  • B
  • C
  • D

Q14:

The figure shows the graph of 𝑦 = π‘₯ s i n .