For any <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>sec</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><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><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>s</mi><mi>e</mi><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> , <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> , to find the vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>2</mn><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> . Set the inside of the secant 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>sec</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"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> to find where the vertical asymptote occurs for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>2</mn><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Set the inside of the secant function <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> equal to <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>2</mn><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> will occur at <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> , where <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><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>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>2</mn><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> occur at <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> , <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> , and every <math><mstyle displaystyle="true"><mi>π</mi><mi>n</mi></mstyle></math> , where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer. This is half of the period.

There are only vertical asymptotes for secant and cosecant functions.

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> for any integer <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math>

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> for any integer <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math>

No Horizontal Asymptotes

No Oblique Asymptotes

Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>sec</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>s</mi><mi>e</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>1</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>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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>1</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</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"><mn>2</mn><mi>π</mi></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><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> for any integer <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math>

Amplitude: None

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

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Name | one billion eight hundred twenty-four million one hundred fifty-four thousand two hundred ninety-six |
---|

- 1824154296 has 64 divisors, whose sum is
**8211611520** - The reverse of 1824154296 is
**6924514281** - Previous prime number is
**3191**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 42
- Digital Root 6

Name | one billion seven hundred fifty-four million five hundred thirty-four thousand six hundred fifty-two |
---|

- 1754534652 has 64 divisors, whose sum is
**4193925120** - The reverse of 1754534652 is
**2564354571** - Previous prime number is
**461**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 42
- Digital Root 6

Name | three hundred twenty million nine hundred thirty-nine thousand four hundred eighteen |
---|

- 320939418 has 16 divisors, whose sum is
**648235296** - The reverse of 320939418 is
**814939023** - Previous prime number is
**101**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 39
- Digital Root 3