Graphing Cotangent

Show your work for each problem.

1.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The period of is , unlike sine and cosine which have period . This is because cotangent completes one full cycle between consecutive asymptotes.
2.
Where are the vertical asymptotes of ?
1. a) for integer
2. b) for integer
3. c) for integer
4. d) for integer
Answer: for integer
Since , it is undefined where , which occurs at for any integer . This is different from tangent, which has asymptotes at .
3.
What is ?
Answer: 1
Since , and , we have .
4.
What is ?
Answer: 0
. This is the x-intercept of the cotangent function.
5.
Does the cotangent function increase or decrease within each period?
1. a)Increases
2. b)First increases, then decreases
3. c)First decreases, then increases
4. d)Decreases
Answer: Decreases
Cotangent decreases from to within each period. This is opposite to tangent, which increases. At the value approaches , passes through 0 at , and approaches as .
6.
What is the period of ? Express your answer as a fraction of (e.g., type 'pi/2' for ).
Answer: pi/3
For , the period is . Here , so the period is . The graph completes 3 cycles in the same interval where the basic cotangent completes 1.
7.
What is the phase shift of ?
1. a)No phase shift
2. b) to the left
3. c) to the right
4. d) to the right
Answer: to the right
In the form , the graph shifts right by . Here , so the graph shifts units to the right. All asymptotes move from to .
8.
Find the period and asymptotes of .
Answer: pi/2, x = n*pi/2
1. What is the coefficient of inside the cotangent? 2
2. Apply the period formula: Period = . What is the period? pi/2
3. The asymptotes occur where . Solve for . x = n*pi/2
9.
Identify the transformations in .
Answer: vertical stretch by 3, vertical shift down 2
1. What does the coefficient 3 in front of cot do? vertical stretch by 3
2. What does the -2 at the end do? vertical shift down 2
3. Does the period change? no
10.
What is the first positive asymptote of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The phase shift of to the right moves all asymptotes from to . For , the asymptote moves from to , which is the first positive asymptote.
11.
For , find the period, phase shift, and the first positive asymptote.
Answer: pi/3, pi/3 right, pi/3
1. Rewrite as . What is the phase shift? pi/3
2. What is the period? (Use formula ) pi/3
3. Basic cotangent has asymptote at . After phase shift of right, where is the first positive asymptote? pi/3
12.
What is the range of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The range of is all real numbers . Adding 5 shifts every output up by 5, but since the range was already all real numbers, adding 5 to every value still gives all real numbers. The range remains .
13.
If has asymptotes at , what is ?
Answer: 4
The asymptotes are at , so the period is . Since Period = , we have , so .
14.
Find the equation of a cotangent function with period , phase shift right, and vertical shift up by 3.
Answer: y = cot(0.5x - pi/4) + 3
1. If period = and Period = , what is ? 0.5
2. For phase shift right with , what is in ? pi/4
3. What is added for a vertical shift of 3 up? 3
15.
Which equation represents a cotangent function with asymptotes at ?
1. a)
2. b)
3. c)
4. d)
Answer:
The asymptotes are spaced apart, so the period is . Since Period = , we have , giving . The first asymptote is at , not 0, but for , asymptotes occur at , i.e., . For , we get , which matches!