Question

worksheet: graphing tangent functions directions: graph each tangent function. show all work and label asymptotes. 1. y = tan(x) (use grid space for graph) 2. y = 2tan(x) (use grid space for graph) 3. y = tan(2x) (use grid space for graph) 4. y = tan(x - π/4) (use grid space for graph) 5. y = -tan(x) (use grid space for graph)

Explanation:

For each tangent function of the form $y = A\tan(B(x-C))$, we first find key features: vertical asymptotes, period, phase shift, and vertical stretch/reflection, then outline the graph.

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1. $y = \tan(x)$

Step1: Identify base tangent parameters

$A=1, B=1, C=0$

Step2: Find vertical asymptotes

Asymptotes at $x = \frac{\pi}{2} + k\pi$, where $k\in\mathbb{Z}$

Step3: Determine period

$\text{Period} = \frac{\pi}{|B|} = \pi$

Step4: Key points for graphing

At $x=0$, $y=0$; curves approach asymptotes $\pm\infty$ near $x=\pm\frac{\pi}{2}$

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2. $y = 2\tan(x)$

Step1: Identify parameters

$A=2, B=1, C=0$

Step2: Find vertical asymptotes

Asymptotes at $x = \frac{\pi}{2} + k\pi$, where $k\in\mathbb{Z}$

Step3: Determine period

$\text{Period} = \frac{\pi}{|B|} = \pi$

Step4: Note vertical stretch

All $y$-values are doubled; at $x=\frac{\pi}{4}$, $y=2$

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3. $y = \tan(2x)$

Step1: Identify parameters

$A=1, B=2, C=0$

Step2: Find vertical asymptotes

Solve $2x = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{4} + \frac{k\pi}{2}$, where $k\in\mathbb{Z}$

Step3: Determine period

$\text{Period} = \frac{\pi}{|B|} = \frac{\pi}{2}$

Step4: Key points for graphing

At $x=0$, $y=0$; at $x=\frac{\pi}{8}$, $y=1$

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4. $y = \tan(x - \pi/4)$

Step1: Identify parameters

$A=1, B=1, C=\frac{\pi}{4}$

Step2: Find vertical asymptotes

Solve $x-\frac{\pi}{4} = \frac{\pi}{2} + k\pi \implies x = \frac{3\pi}{4} + k\pi$, where $k\in\mathbb{Z}$

Step3: Determine period

$\text{Period} = \frac{\pi}{|B|} = \pi$

Step4: Note phase shift

Graph shifts right by $\frac{\pi}{4}$; at $x=\frac{\pi}{4}$, $y=0$

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5. $y = -\tan(x)$

Step1: Identify parameters

$A=-1, B=1, C=0$

Step2: Find vertical asymptotes

Asymptotes at $x = \frac{\pi}{2} + k\pi$, where $k\in\mathbb{Z}$

Step3: Determine period

$\text{Period} = \frac{\pi}{|B|} = \pi$

Step4: Note reflection

Graph is flipped over x-axis; at $x=\frac{\pi}{4}$, $y=-1$

Answer:

1. $y=\tan(x)$: Asymptotes at $x=\frac{\pi}{2}+k\pi$ ($k\in\mathbb{Z}$), period $\pi$, passes through $(0,0)$, increases to $+\infty$ left of $\frac{\pi}{2}$ and $-\infty$ right of $-\frac{\pi}{2}$.
2. $y=2\tan(x)$: Asymptotes at $x=\frac{\pi}{2}+k\pi$ ($k\in\mathbb{Z}$), period $\pi$, vertical stretch by 2, passes through $(0,0)$, at $x=\frac{\pi}{4}$ $y=2$.
3. $y=\tan(2x)$: Asymptotes at $x=\frac{\pi}{4}+\frac{k\pi}{2}$ ($k\in\mathbb{Z}$), period $\frac{\pi}{2}$, passes through $(0,0)$, at $x=\frac{\pi}{8}$ $y=1$.
4. $y=\tan(x-\pi/4)$: Asymptotes at $x=\frac{3\pi}{4}+k\pi$ ($k\in\mathbb{Z}$), period $\pi$, right shift by $\frac{\pi}{4}$, passes through $(\frac{\pi}{4},0)$.
5. $y=-\tan(x)$: Asymptotes at $x=\frac{\pi}{2}+k\pi$ ($k\in\mathbb{Z}$), period $\pi$, reflection over x-axis, passes through $(0,0)$, at $x=\frac{\pi}{4}$ $y=-1$.

(Graph each function on grid paper using the asymptotes, period, and key points above, labeling all asymptotes with their equations.)