Graphing the Sine Function

Show your work for each problem.

1.
What is the amplitude of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The basic sine function oscillates between and , so the amplitude (distance from midline 0 to peak 1) is .
2.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The sine function completes one full cycle from 0 to radians (or 360 degrees). In radians, the period is .
3.
What is ?
Answer: 0
. The sine curve passes through the origin, starting its wave pattern from zero.
4.
What is ?
Answer: 1
. At radians (90 degrees), the sine reaches its maximum value.
5.
What is the range of ?
1. a)All real numbers
2. b)
3. c)
4. d)
Answer:
The sine function oscillates between a minimum of and a maximum of , so the range is .
6.
What is the amplitude of ?
Answer: 5
In , the amplitude is . Here , so the amplitude is .
7.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
Period . The graph completes one cycle in radians.
8.
What is the period of ? Express your answer in terms of .
Answer: pi/2
Period . The coefficient 4 compresses the graph horizontally.
9.
What is the range of ?
1. a)
2. b)
3. c)
4. d)
Answer:
With amplitude 3 and midline at , the range is from to , giving .
10.
Find the amplitude and period of .
Answer: amplitude = 2, period = 2pi/3
1. What is the coefficient in front of sine (the value of A)? 2
2. What is the amplitude? 2
3. What is the coefficient of x inside sine (the value of B)? 3
4. What is the period? (Use the formula and simplify) 2pi/3
11.
In which direction does shift compared to ?
1. a)Up by
2. b)Right by
3. c)Left by
4. d)Down by
Answer: Right by
When we have inside sine, the graph shifts RIGHT by . Here , so it shifts right by .
12.
Find the range of .
Answer: [1, 5]
1. What is the amplitude of this function? 2
2. What is the vertical shift (midline)? 3
3. What is the minimum value (midline - amplitude)? 1
4. What is the maximum value (midline + amplitude)? 5
13.
For , find the amplitude, period, phase shift, and vertical shift.
Answer: amplitude = 3, period = pi, phase shift = pi/4 right, vertical shift = 1 up
1. What is the amplitude (|A|)? 3
2. What is B (coefficient of x)? 2
3. What is the period (2π/|B|)? pi
4. What is C (the value being subtracted from Bx)? pi/2
5. What is the phase shift (C/B)? Include direction. pi/4 right
6. What is the vertical shift (D)? 1
14.
What is the range of ?
1. a)
2. b)
3. c)
4. d)
Answer:
Amplitude = 2 (the negative reflects the graph but doesn't change range). Midline = 4. Range: .
15.
A sound wave has a frequency of 440 Hz. What is its period in seconds? Round to 4 decimal places.
Answer: 0.0023
Period seconds, which rounds to seconds.
16.
Write the equation for a sine function with amplitude 4, period , phase shift to the right, and midline .
Answer: y = 4sin(2x - pi/3) - 2
1. What is A (the amplitude)? 4
2. If period = π, what is B? (Use 2π/B = π) 2
3. Phase shift = C/B = π/6. With B=2, what is C? pi/3
4. What is D (the vertical shift)? -2
5. Write the complete equation y = 4sin(2x - pi/3) - 2