Graphing the Tangent Function

Show your work for each problem.

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