Graphing the Cosine Function

Show your work for each problem.

1.
What is the value of ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This is the maximum value of the cosine function, which occurs at the starting point.
2.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The period of is because it takes radians (one full rotation around the unit circle) for the function to complete one cycle.
3.
What is the amplitude of ?
Answer: 1
The amplitude of is . The function oscillates between and , so the distance from the midline (at ) to the maximum is .
4.
What is the value of ?
Answer: -1
. At radians (180 degrees), we are at the leftmost point of the unit circle, which gives the minimum value of cosine.
5.
At what x-value does first cross the x-axis (starting from )?
1. a)
2. b)
3. c)
4. d)
Answer:
The cosine function first crosses the x-axis at . At this point, .
6.
What is the amplitude of ?
Answer: 5
For , the amplitude is . Here, , so the amplitude is . The function oscillates between and .
7.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
For , the period is . Here, , so the period is .
8.
What is the period of ? Express your answer as a fraction of (e.g., enter "pi/2" for ).
Answer: pi/2
Period . The function completes 4 cycles in the space where completes one.
9.
The graph of is shifted in which direction compared to ?
1. a)Right by
2. b)Down by
3. c)Left by
4. d)Up by
Answer: Right by
For , the graph shifts RIGHT by units. Here , so the shift is units to the right.
10.
What is the maximum value of ?
Answer: 5
Maximum of is . So maximum of .
11.
Find the amplitude, period, and midline of .
Answer: 4, pi, 1
1. Identify A (the coefficient in front of cosine). What is the amplitude? 4
2. Identify B (the coefficient of x inside cosine). What is B? 2
3. Calculate the period using . What is the period? pi
4. Identify D (the number added at the end). What is the midline? 1
12.
Describe all transformations of compared to .
Answer: reflection, stretch by 3, shift up 2
1. What does the negative sign in front of the 3 do? reflection
2. What is the amplitude (ignoring the sign)? 3
3. What does the +2 do to the graph? shift up 2
13.
Find the equation of a cosine function with amplitude 2, period , and phase shift to the right.
Answer: y = 2cos(2(x - pi/6))
1. What is A (the amplitude)? 2
2. If the period is , what is B? Use period = 2
3. For a phase shift of to the right, what goes inside the cosine? x - pi/6
4. Combine into the full equation: 2cos(2(x - pi/6))
14.
Which equation represents a cosine function with a maximum at and minimum at ?
1. a)
2. b)
3. c)
4. d)
Answer:
Amplitude . Midline . Since max is at , no phase shift needed. Answer: .
15.
A ferris wheel has a diameter of 40 meters and its center is 25 meters above the ground. It takes 2 minutes for one full rotation. Write a cosine function for a rider's height above the ground, where is in minutes and the rider starts at the top.
Answer: h(t) = 20cos(pi*t) + 25
1. What is the amplitude (radius of the wheel)? 20
2. What is the vertical shift (height of the center)? 25
3. If the period is 2 minutes, what is B? pi
4. Write the final equation 20cos(pi*t) + 25
16.
What is the minimum value of ?
Answer: -2
Since the cosine is reflected (negative), the minimum occurs when : .