Graphs of tangent and cotangent functions have repeating branches separated by vertical asymptotes. Tangent graphs have asymptotes where cosine is zero, while cotangent graphs have asymptotes where sine is zero. These problems include vertical stretches, reflections, period changes, horizontal shifts, vertical shifts, and identifying important features of tangent and cotangent graphs.

Practice Problems

\(\textbf{1)}\) \(f(x)=4 \tan{(x)}\)

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\(f(x)=4\tan(x)\)

\(\text{The tangent graph has period }\pi.\)

\(\text{The factor }4\text{ creates a vertical stretch by }4.\)

\(\text{Vertical asymptotes occur at }x=\frac{\pi}{2}+k\pi.\)

\(\text{The midline is }y=0.\)

 

\(\textbf{2)}\) \(f(x)=-\tan{(\pi x)}\)

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\(f(x)=-\tan(\pi x)\)

\(\text{For }f(x)=a\tan(bx),\text{ the period is }\frac{\pi}{|b|}.\)

\(\text{Here }b=\pi,\text{ so the period is }\frac{\pi}{\pi}=1.\)

\(\text{The negative sign reflects the graph over the }x\text{-axis.}\)

\(\text{Vertical asymptotes occur at }x=\frac{1}{2}+k.\)

 

\(\textbf{3)}\) \(f(x)=\cot{(2x)}\)

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\(f(x)=\cot(2x)\)

\(\text{For }f(x)=a\cot(bx),\text{ the period is }\frac{\pi}{|b|}.\)

\(\text{Here }b=2,\text{ so the period is }\frac{\pi}{2}.\)

\(\text{Cotangent has vertical asymptotes where }2x=k\pi.\)

\(\text{So vertical asymptotes occur at }x=\frac{k\pi}{2}.\)

 

\(\textbf{4)}\) \(f(x)=\cot{\left(\frac{x}{2}\right)}\)

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\(f(x)=\cot\left(\frac{x}{2}\right)\)

\(\text{Here }b=\frac{1}{2}.\)

\(\text{The period is }\frac{\pi}{\left|\frac{1}{2}\right|}=2\pi.\)

\(\text{Cotangent has vertical asymptotes where }\frac{x}{2}=k\pi.\)

\(\text{So vertical asymptotes occur at }x=2k\pi.\)

 

\(\textbf{5)}\) \(f(x)=\tan{\left(x+\frac{\pi}{2}\right)}\)

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\(f(x)=\tan\left(x+\frac{\pi}{2}\right)\)

\(\text{This is the parent tangent graph shifted left }\frac{\pi}{2}.\)

\(\text{The period is }\pi.\)

\(\text{Vertical asymptotes occur when }x+\frac{\pi}{2}=\frac{\pi}{2}+k\pi.\)

\(\text{So vertical asymptotes occur at }x=k\pi.\)

 

\(\textbf{6)}\) \(f(x)=3 + \tan{(x)}\)

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\(f(x)=3+\tan(x)\)

\(\text{This is the parent tangent graph shifted up }3.\)

\(\text{The period is }\pi.\)

\(\text{The midline is }y=3.\)

\(\text{Vertical asymptotes occur at }x=\frac{\pi}{2}+k\pi.\)

 

\(\textbf{7)}\) \(f(x)=-\cot{(x)}-2\)

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\(f(x)=-\cot(x)-2\)

\(\text{The negative sign reflects the cotangent graph over the }x\text{-axis.}\)

\(\text{The }-2\text{ shifts the graph down }2.\)

\(\text{The period is }\pi.\)

\(\text{Vertical asymptotes occur at }x=k\pi.\)

 

\(\textbf{8)}\) \(f(x)=3\tan{(2x)}-4\)

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\(f(x)=3\tan(2x)-4\)

\(\text{Here }a=3,\ b=2,\text{ and }d=-4.\)

\(\text{The period is }\frac{\pi}{2}.\)

\(\text{The graph is vertically stretched by }3\text{ and shifted down }4.\)

\(\text{The midline is }y=-4.\)

\(\text{Vertical asymptotes occur where }2x=\frac{\pi}{2}+k\pi.\)

\(\text{So }x=\frac{\pi}{4}+\frac{k\pi}{2}.\)

 

\(\textbf{9)}\) \(f(x)=-\cot{(\pi x)}\)

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\(f(x)=-\cot(\pi x)\)

\(\text{Here }b=\pi.\)

\(\text{The period is }\frac{\pi}{\pi}=1.\)

\(\text{The negative sign reflects the cotangent graph over the }x\text{-axis.}\)

\(\text{Vertical asymptotes occur where }\pi x=k\pi.\)

\(\text{So vertical asymptotes occur at }x=k.\)

 

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