Test Objectives
• Demonstrate the ability to sketch the graphs of tangent and cotangent
• Demonstrate the ability to find the period of the tangent or cotangent function
• Demonstrate the ability to find the phase shift of the tangent or cotangent function
• Demonstrate the ability to find the vertical shift of the tangent or cotangent function
• Demonstrate the ability to find a pair of consecutive vertical asymptotes for the tangent or cotangent function
Graphing Tangent and Cotangent Practice Test:

#1:

Instructions: Find the period, phase shift, vertical shift, and a pair of consecutive vertical asymptotes. Sketch the graph.

$$a)\hspace{.1em}f(x)=2\tan\left(\frac{x}{2}- \frac{5π}{6}\right) - 2$$

$$b)\hspace{.1em}f(x)=3\tan\left(2x + \frac{5π}{3}\right) - 2$$

#2:

Instructions: Find the period, phase shift, vertical shift, and a pair of consecutive vertical asymptotes. Sketch the graph.

$$a)\hspace{.1em}f(x)=4\tan\left(2x + \frac{π}{3}\right) + 2$$

$$b)\hspace{.1em}f(x)={-}\frac{1}{2}\tan\left(x + \frac{π}{2}\right)$$

#3:

Instructions: Find the period, phase shift, vertical shift, and a pair of consecutive vertical asymptotes. Sketch the graph.

$$a)\hspace{.1em}f(x)=\frac{1}{4}\tan\left(\frac{x}{4}- \frac{2π}{3}\right) + 1$$

$$b)\hspace{.1em}f(x)=3\cot\left(\frac{x}{3}- \frac{5π}{6}\right) + 2$$

#4:

Instructions: Find the period, phase shift, vertical shift, and a pair of consecutive vertical asymptotes. Sketch the graph.

$$a)\hspace{.1em}f(x)=2\cot\left(\frac{x}{2}+ \frac{5π}{4}\right) - 2$$

$$b)\hspace{.1em}f(x)=3\cot\left(\frac{x}{3}+ \frac{π}{3}\right) + 2$$

#5:

Instructions: Find the period, phase shift, vertical shift, and a pair of consecutive vertical asymptotes. Sketch the graph.

$$a)\hspace{.1em}f(x)=3\cot\left(2x - \frac{5π}{6}\right) + 1$$

$$b)\hspace{.1em}f(x)={-}\cot\left(x - \frac{π}{3}\right)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\text{Period:}\hspace{.1em}2π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{5π}{3}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Down}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=\frac{2π}{3}$$ $$\text{and}$$ $$x=\frac{8π}{3}$$ Formula for Vertical Asymptotes: $$x=\frac{2π}{3}+ 2πn$$ Where n is any integer. Desmos link for more detail

$$b)\hspace{.1em}\text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{5π}{6}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Down}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=-\frac{13π}{12}$$ $$\text{and}$$ $$x=-\frac{7π}{12}$$ Formula for Vertical Asymptotes: $$x=-\frac{13π}{12}+ \frac{πn}{2}$$ Where n is any integer. Desmos link for more detail

#2:

Solutions:

$$a)\hspace{.1em}\text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{π}{6}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=-\frac{5π}{12}$$ $$\text{and}$$ $$x=\frac{π}{12}$$ Formula for Vertical Asymptotes: $$x=-\frac{5π}{12}+ \frac{πn}{2}$$ Where n is any integer. Desmos link for more detail

$$b)\hspace{.1em}\text{Period:}\hspace{.1em}π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{π}{2}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{None}$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=-π$$ $$\text{and}$$ $$x=0$$ Formula for Vertical Asymptotes: $$x=-π + πn$$ Where n is any integer. Desmos link for more detail

#3:

Solutions:

$$a)\hspace{.1em}\text{Period:}\hspace{.1em}4π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{8π}{3}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}1$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=\frac{2π}{3}$$ $$\text{and}$$ $$x=\frac{14π}{3}$$ Formula for Vertical Asymptotes: $$x=\frac{2π}{3}+ 4πn$$ Where n is any integer. Desmos link for more detail

$$b)\hspace{.1em}\text{Period:}\hspace{.1em}3π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{5π}{2}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=\frac{5π}{2}$$ $$\text{and}$$ $$x=\frac{11π}{2}$$ Formula for Vertical Asymptotes: $$x=\frac{5π}{2}+ 3πn$$ Where n is any integer. Desmos link for more detail

#4:

Solutions:

$$a)\hspace{.1em}\text{Period:}\hspace{.1em}2π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{5π}{2}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Down}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=-\frac{5π}{2}$$ $$\text{and}$$ $$x=-\frac{π}{2}$$ Formula for Vertical Asymptotes: $$x=-\frac{5π}{2}+ 2πn$$ Where n is any integer. Desmos link for more detail

$$b)\hspace{.1em}\text{Period:}\hspace{.1em}3π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}π$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}2$$ $$\text{Vertical Asymptotes:}\hspace{.1em}$$ $$x=-π$$ $$\text{and}$$ $$x=2π$$ Formula for Vertical Asymptotes: $$x=-π + 3πn$$ Where n is any integer. Desmos link for more detail

#5:

Solutions:

$$a)\hspace{.1em}\text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{5π}{12}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}1$$ $$\text{Vertical Asymptotes:}$$ $$x=\frac{5π}{12}$$ $$\text{and}$$ $$x=\frac{11π}{12}$$ Formula for Vertical Asymptotes: $$x=\frac{5π}{12}+ \frac{πn}{2}$$ Where n is any integer. Desmos link for more detail

$$b)\hspace{.1em}\text{Period:}\hspace{.1em}π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{π}{3}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{None}\hspace{.1em}$$ $$\text{Vertical Asymptotes:}$$ $$x=\frac{π}{3}$$ $$\text{and}$$ $$x=\frac{4π}{3}$$ Formula for Vertical Asymptotes: $$x=\frac{π}{3}+ πn$$ Where n is any integer. Desmos link for more detail