Graphing Trigonometric Functions 3

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Question 1 of 4

1. Question
Graph the trigonometric function within the domain $- \frac{π}{4} \leq x \leq \frac{π}{4}$ $-pi/4≤x≤pi/4$

$y = 2 \tan 4 x$ $y=2tan 4x$
- 1.
- 2.
- 3.
- 4.
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Period Fomula

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

First, solve for the period of the function

$b$ $b$ $=$ $=$ $4$ $4$

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

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

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

$tan - \frac{π}{8}$ $\text(tan) -pi/8$ $=$ $=$ $undefined$ $\text(undefined)$

$tan \frac{π}{8}$ $\text(tan) pi/8$ $=$ $=$ $undefined$ $\text(undefined)$

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

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

First curve

Second curve

Third curve

Question 2 of 4

Graph the trigonometric function within the domain

- \frac{π}{2} \leq x \leq 2 π

$-pi/2≤x≤2pi$

y = \cot x

$y=cot x$

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Period Fomula

P_{cot} = \frac{π}{b}

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

First, solve for the period of the function

b

$b$

=

$=$

1

$1$

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

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

	$=$ $=$	$π$ $pi$

Recall that

cot = \frac{1}{tan}

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

Asymptotes will occur within the domain for the values of

\cot

$cot$ that are undefined or when the value of

\tan

$tan$ is

0

$0$

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

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

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

To graph the

\cot

$cot$ curve, start at x-axis of the lower domain

(- \frac{π}{2})

$(-pi/2)$ 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

(2 π)

$(2pi)$

First curve

Second curve

Third curve

Question 3 of 4

Graph the trigonometric function within the domain

- 1 \leq x \leq 1

$-1≤x≤1$

y = \cot π x

$y=cot pix$

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Period Fomula

P_{cot} = \frac{π}{b}

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

First, solve for the period of the function

b

$b$

=

$=$

π

$pi$

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

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

	$=$ $=$	$1$ $1$

Recall that

cot = \frac{1}{tan}

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

Asymptotes will occur within the domain for the values of

\cot

$cot$ that are undefined or when the value of

\tan

$tan$ is

0

$0$

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

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

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

To graph the

\cot

$cot$ curve, start at the lower domain

(- 1)

$(-1)$ 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)

$(1)$

First curve

Second curve

Question 4 of 4

4. Question
Graph the trigonometric function

$y = \sin (x + \frac{π}{4})$ $y=sin(x+pi/4)$
- 1.
- 2.
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- 4.
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General Form of a Sin Function with Horizontal Shift

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

Period Fomula

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

First, identify the values of the function

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

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

$a$ $a$ $=$ $=$ $1$ $1$

$b$ $b$ $=$ $=$ $1$ $1$

$h$ $h$ $=$ $=$ $\frac{π}{4}$ $pi/4$

Next, solve for the period of the function

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

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

$=$ $=$ $2 π$ $2pi$

To graph the $\sin$ $sin$ 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 $(0, 0)$ $(0,0)$

Curve reaches peak of amplitude ( $a$ $a$ ) 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 ( $P = 4 π$ $P=4pi$ )

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

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

Graphing 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

$tan - \frac{π}{8}$ $\text(tan) -pi/8$	$=$ $=$	$undefined$ $\text(undefined)$

$tan \frac{π}{8}$ $\text(tan) pi/8$	$=$ $=$	$undefined$ $\text(undefined)$

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

$a$ $a$	$=$ $=$	$1$ $1$
$b$ $b$	$=$ $=$	$1$ $1$

$h$ $h$	$=$ $=$	$\frac{π}{4}$ $pi/4$

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

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

	$=$ $=$	$2 π$ $2pi$