Graph Trigonometric Functions 3

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Question 1 of 4

1. Question
Graph the trigonometric function within the domain $-pi/4≤x≤pi/4$

$y=2tan 4x$
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Period Fomula

$P_{\text{tan}}=\frac{\pi}{\color{#00880A}{b}}$

First, solve for the period of the function

$b$ $=$ $4$

$P_{\text{tan}}$ $=$ $\frac{\pi}{\color{#00880A}{b}}$

$=$ $\frac{\pi}{\color{#00880A}{4}}$ Substitute known values

Asymptotes will occur within the domain for the values of $tan$ that are undefined

$\text(tan) -pi/8$ $=$ $\text(undefined)$

$\text(tan) pi/8$ $=$ $\text(undefined)$

To graph the $tan$ curve, start at x-axis of the lower domain $(-pi/4)$ and draw a curve heading up to the first asymptote $(-pi/8)$ . This will cover half of a period of the curve.

Repeat the process until you reach the x-axis of the upper domain $(pi/4)$ , which will also show half of the curve.

First curve

Second curve

Third curve
Question 2 of 4

2. Question
Graph the trigonometric function within the domain $-pi/2≤x≤2pi$

$y=cot x$
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Period Fomula

$P_{\text{cot}}=\frac{\pi}{\color{#00880A}{b}}$

First, solve for the period of the function

$b$ $=$ $1$

$P_{\text{tan}}$ $=$ $\frac{\pi}{\color{#00880A}{b}}$

$=$ $\frac{\pi}{\color{#00880A}{1}}$ Substitute known values

$=$ $pi$

Recall that $\text(cot)=1/(\text(tan))$

Asymptotes will occur within the domain for the values of $cot$ that are undefined or when the value of $tan$ is $0$

$\text(cot) 0$ $=$ $\text(undefined)$

$\text(cot) pi$ $=$ $\text(undefined)$

$\text(cot) 2pi$ $=$ $\text(undefined)$

To graph the $cot$ curve, start at x-axis of the lower domain $(-pi/2)$ and draw a curve and heading down to the first asymptote at the y-axis. This will cover half of a period of the curve.

Repeat the process until you reach the x-axis of the upper domain $(2pi)$

First curve

Second curve

Third curve
Question 3 of 4

3. Question
Graph the trigonometric function within the domain $-1≤x≤1$

$y=cot pix$
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Period Fomula

$P_{\text{cot}}=\frac{\pi}{\color{#00880A}{b}}$

First, solve for the period of the function

$b$ $=$ $pi$

$P_{\text{tan}}$ $=$ $\frac{\pi}{\color{#00880A}{b}}$

$=$ $\frac{\pi}{\color{#00880A}{pi}}$ Substitute known values

$=$ $1$

Recall that $\text(cot)=1/(\text(tan))$

Asymptotes will occur within the domain for the values of $cot$ that are undefined or when the value of $tan$ is $0$

$\text(cot) -1$ $=$ $\text(undefined)$

$\text(cot) 0$ $=$ $\text(undefined)$

$\text(cot) 1$ $=$ $\text(undefined)$

To graph the $cot$ curve, start at the lower domain $(-1)$ and draw a curve heading down to the first asymptote at the y-axis. This will cover a period of the curve.

Repeat the process until you reach the x-axis of the upper domain $(1)$

First curve

Second curve
Question 4 of 4

4. Question
Graph the trigonometric function

$y=sin(x+pi/4)$
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General Form of a Sin Function with Horizontal Shift

$y=$ $a$ $\text(sin)$ $b$ $(x-$ $h$ $)$

Period Fomula

$P=\frac{2\pi}{\color{#00880A}{b}}$

First, identify the values of the function

$y$ $=$ $a$ $\text(sin)$ $b$ $(x-$ $h$ $)$

$y$ $=$ $\color{#004ec4}{1}\;\text{sin}\;\color{#00880A}{1}(x+\color{#9a00c7}{\frac{\pi}{4}})$

$a$ $=$ $1$

$b$ $=$ $1$

$h$ $=$ $pi/4$

Next, solve for the period of the function

$P$ $=$ $\frac{2\pi}{\color{#00880A}{b}}$

$=$ $\frac{2\pi}{\color{#00880A}{1}}$ Substitute known values

$=$ $2pi$

To graph the $sin$ curve, it is better to divide the value of its period into for parts and have the curve meet the following conditions

Curve starts at $(0,0)$

Curve reaches peak of amplitude ( $a$ ) at 1st quarter

Curve intercepts x-axis at 2nd quarter

Curve reaches minimum amplitude at 3rd quarter

Curve starts at x-axis again at the period ( $P=4pi$ )

This will be the sin curve for $y=\text(sin) x+pi/4$ with a period of $2pi$

Shift the graph horizontally $h=pi/4$ units to the left. Since the labels are in intervals of $pi/4$ , we move the graph $1$ label to the left

Graph Trigonometric Functions 3

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Quiz summary

Information

1. Question

Hint

Excellent!

Period Fomula

2. Question

Hint

Fantastic!

Period Fomula

3. Question

Hint

Keep Going!

Period Fomula

4. Question

Hint

Correct!

General Form of a Sin Function with Horizontal Shift

Period Fomula

Quizzes

$P_{\text{tan}}$	$=$	$\frac{\pi}{\color{#00880A}{b}}$

	$=$	$\frac{\pi}{\color{#00880A}{4}}$	Substitute known values

$\text(tan) -pi/8$	$=$	$\text(undefined)$

$\text(tan) pi/8$	$=$	$\text(undefined)$

$P_{\text{tan}}$	$=$	$\frac{\pi}{\color{#00880A}{b}}$

	$=$	$\frac{\pi}{\color{#00880A}{1}}$	Substitute known values

	$=$	$pi$

$\text(cot) 0$	$=$	$\text(undefined)$

$\text(cot) pi$	$=$	$\text(undefined)$

$\text(cot) 2pi$	$=$	$\text(undefined)$

$P_{\text{tan}}$	$=$	$\frac{\pi}{\color{#00880A}{b}}$

	$=$	$\frac{\pi}{\color{#00880A}{pi}}$	Substitute known values

	$=$	$1$

$\text(cot) -1$	$=$	$\text(undefined)$

$\text(cot) 0$	$=$	$\text(undefined)$

$\text(cot) 1$	$=$	$\text(undefined)$

$y$	$=$	$a$ $\text(sin)$ $b$ $(x-$ $h$ $)$
$y$	$=$	$\color{#004ec4}{1}\;\text{sin}\;\color{#00880A}{1}(x+\color{#9a00c7}{\frac{\pi}{4}})$

$P$	$=$	$\frac{2\pi}{\color{#00880A}{b}}$

	$=$	$\frac{2\pi}{\color{#00880A}{1}}$	Substitute known values

	$=$	$2pi$