Graphing the Tangent Function

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 , not like sine and cosine. This is because for all in the domain.
2.
Where does the first positive asymptote of occur?
1. a)
2. b)
3. c)
4. d)
Answer:
The asymptotes of occur where . The first positive value where this happens is .
3.
What is ?
Answer: 0
. The tangent function passes through the origin.
4.
What is the range of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The range of is all real numbers . Unlike sine and cosine which are bounded between and , tangent can take any real value.
5.
What is ?
Answer: 1
. This is a key reference point on the tangent graph.
6.
What is the period of ? Give your answer as a fraction of (e.g., enter 'pi/2' for ).
Answer: pi/2
For , the period is . With , the period is .
7.
Which transformation does the '3' represent in ?
1. a)Vertical stretch by factor of 3
2. b)Vertical shift up by 3
3. c)Horizontal compression by factor of 3
4. d)Period change to
Answer: Vertical stretch by factor of 3
The coefficient 3 in front of stretches the graph vertically by a factor of 3. Every y-value is multiplied by 3, making the curve steeper.
8.
Find the period and the first positive asymptote of .
Answer: pi/6
1. What is the formula for the period of ? pi/|b|
2. What is in ? 3
3. Calculate the period: . Write as fraction of pi. pi/3
4. The first positive asymptote is at half the period. What is ? pi/6
9.
What happens to the asymptotes when we graph ?
1. a)They shift right by
2. b)They don't change position
3. c)They shift down by
4. d)They shift left by
Answer: They shift right by
The transformation shifts the entire graph, including asymptotes, to the right by . The new asymptotes are at , etc.
10.
Find where equals 1 for in .
Answer: pi/4, 5pi/4
1. At what angle in the first quadrant does ? pi/4
2. The period of tangent is . What is ? 5pi/4
3. Is less than ? yes
4. List both solutions in : pi/4, 5pi/4
11.
What is the period of ? Give your answer as a multiple of (e.g., '2pi' for ).
Answer: 2pi
Period = . The graph is stretched horizontally, taking twice as long to complete one cycle.
12.
Graph . Describe how it differs from .
Answer: reflected over x-axis
1. What does the negative sign in front of a function do? reflection
2. At , what is ? -1
3. Do the asymptotes change position? no
4. Describe the transformation in one phrase: reflected over x-axis
13.
For , what is the equation of the midline?
1. a)
2. b)There is no midline
3. c)
4. d)
Answer:
The vertical shift of moves the entire graph up by 2 units. The midline (where the function crosses between periods) is at .
14.
Find the asymptotes of in the interval .
Answer: pi/4, 3pi/4
1. Rewrite as . What is the phase shift? pi/2
2. For , asymptotes are at . With a shift of , where is the first asymptote? pi/4
3. The period is . What is the next asymptote after ? 3pi/4
4. List the asymptotes in : pi/4, 3pi/4
15.
Which statement is TRUE about the graph of ?
1. a)It has a maximum value of 1
2. b)It is an odd function:
3. c)It is continuous for all real numbers
4. d)It is an even function:
Answer: It is an odd function:
The tangent function is odd because . The graph has point symmetry about the origin.