For any <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> , vertical asymptotes occur at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi></mstyle></math> , where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer. Use the basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>c</mi><mi>s</mi><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> , <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>)</mo></mrow></mstyle></math> , to find the vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>4</mn><mi>csc</mi><mrow><mo>(</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> . Set the inside of the cosecant function, <math><mstyle displaystyle="true"><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> , for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>a</mi><mi>csc</mi><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the vertical asymptote occurs for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>4</mn><mi>csc</mi><mrow><mo>(</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>4</mn><mi>x</mi><mo>=</mo><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Set the inside of the cosecant function <math><mstyle displaystyle="true"><mn>4</mn><mi>x</mi></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>4</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>4</mn><mi>csc</mi><mrow><mo>(</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> will occur at <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> , where <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> are vertical asymptotes.

Find the period <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>4</mn><mi>csc</mi><mrow><mo>(</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> occur at <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> , and every <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> , where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>csc</mi><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>-</mo><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi></mstyle></math> to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Since the graph of the function <math><mstyle displaystyle="true"><mi>c</mi><mi>s</mi><mi>c</mi></mstyle></math> does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> in the formula for period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The phase shift of the function can be calculated from <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>

Replace the values of <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> in the equation for phase shift.

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mn>0</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Find the vertical shift <math><mstyle displaystyle="true"><mi>d</mi></mstyle></math> .

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: None

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Amplitude: None

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Do you know how to Graph y=4csc(4x)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one hundred million two hundred eighty-one thousand five hundred fifty-six |
---|

- 100281556 has 8 divisors, whose sum is
**225633510** - The reverse of 100281556 is
**655182001** - Previous prime number is
**2**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 28
- Digital Root 1

Name | eight hundred thirty-five million three hundred sixty-eight thousand one hundred forty-seven |
---|

- 835368147 has 32 divisors, whose sum is
**1544696832** - The reverse of 835368147 is
**741863538** - Previous prime number is
**347**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 45
- Digital Root 9

Name | four hundred forty-nine million six hundred eighty thousand eight hundred eighty-seven |
---|

- 449680887 has 16 divisors, whose sum is
**801665280** - The reverse of 449680887 is
**788086944** - Previous prime number is
**359**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 54
- Digital Root 9