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Back to Graphs of sin, cos, tan (amplitude, period, phase shift)

Graphs of Trigonometric Functions

Grade 10·21 problems·~35 min·ACT Math·topic·act-geo-trig-graphs
Work through problems with immediate feedback
A

Recall / Warm-Up

1.

What is the value of sin⁡ ⁣(π2)\sin\!\left(\frac{\pi}{2}\right)?

2.

What is the value of cos⁡(π)\cos(\pi)?

3.

How many radians are in a full rotation around the unit circle?

B

Fluency Practice

1.

What is the amplitude of y=4sin⁡(x)y = 4\sin(x)?

2.

What is the period of y=sin⁡(3x)y = \sin(3x)?

3.

What is the period of y=cos⁡(4x)y = \cos(4x)? Express your answer in terms of π\pi.

4.

What is the phase shift of y=sin⁡(2x−π)y = \sin(2x - \pi)?

5.

What are the amplitude and midline of y=3cos⁡(x)+5y = 3\cos(x) + 5?

C

Varied Practice

1.

Which of the following is the period of y=tan⁡(2x)y = \tan(2x)?

2.

For the function y=2sin⁡(3x−π)+4y = 2\sin(3x - \pi) + 4:

Amplitude =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

Period =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

Phase shift =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   (express as a fraction of π\pi; use "right" or "left")

Midline: y=y =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

amplitude:
period:
phase shift:
midline:
3.

Which transformation most directly maps y=sin⁡(x)y = \sin(x) onto y=−sin⁡(x)y = -\sin(x)?

4.

A sinusoidal function has a maximum value of 7 and a minimum value of 1. What is the amplitude?

5.

A sinusoidal function has a maximum value of 7 and a minimum value of 1. What is the equation of the midline?

D

Word Problems / Application

1.

A Ferris wheel has a radius of 20 meters and its center is 25 meters above the ground. A rider's height above the ground is modeled by y=−20cos⁡ ⁣(π30t)+25y = -20\cos\!\left(\frac{\pi}{30}t\right) + 25, where tt is time in seconds.

What is the period of the Ferris wheel (the time for one full rotation)?

2.

A sinusoidal graph shows a wave with maximum yy-value of 5 at x=0x = 0, minimum yy-value of −1-1 at x=πx = \pi, and the pattern repeats.

1.

What is the amplitude of this function?

2.

Which equation matches this graph?

3.

A sinusoidal graph has a maximum of 6, a minimum of 2, and the first upward crossing of the midline occurs at x=π6x = \frac{\pi}{6}. The period is 2π3\frac{2\pi}{3}.

Which equation matches this graph?

E

Error Analysis

1.

Taylor solved this problem:

"Find the period and amplitude of y=sin⁡(3x)y = \sin(3x)."

Taylor's work:

  1. Amplitude =3= 3
  2. Period =2π= 2\pi

What error did Taylor make?

2.

Jordan solved this problem:

"Find the phase shift of y=sin⁡(2x+π)y = \sin(2x + \pi)."

Jordan's work:

  1. The phase shift is −π-\pi (opposite sign of CC).
  2. The graph shifts π\pi units to the left.

What error did Jordan make, and what is the correct phase shift?

F

Challenge / Extension

1.

Which of the following is equivalent to y=−cos⁡(x)y = -\cos(x)?

2.

A student claims that the graph of y=3sin⁡ ⁣(2x−π2)+1y = 3\sin\!\left(2x - \frac{\pi}{2}\right) + 1 is identical to the graph of y=−3cos⁡(2x)+1y = -3\cos(2x) + 1. Is the student correct? Justify your answer by identifying the key features (amplitude, period, phase shift, midline) of each function.

0 of 21 answered