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`
Hint

Help Video

Correct

Excellent!

Incorrect

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`
Hint

Help Video

Correct

Fantastic!

Incorrect

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`
Hint

Help Video

Correct

Keep Going!

Incorrect

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)`
Hint

Help Video

Correct

Correct!

Incorrect

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`