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>1</mn><mo>+</mo><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn><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>1</mn><mo>+</mo><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn><mo>)</mo></mrow></mstyle></math> .

Solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>135</mn></mstyle></math> to both sides of the equation.

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify both sides of the equation.

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

Cancel the common factor.

Rewrite the expression.

Simplify <math><mstyle displaystyle="true"><mn>2</mn><mo>⋅</mo><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>135</mn><mo>)</mo></mrow></mstyle></math> .

Apply the distributive property.

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

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> into the numerator.

Cancel the common factor.

Rewrite the expression.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>135</mn></mstyle></math> .

Set the inside of the secant function <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn></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> .

Solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>135</mn></mstyle></math> to both sides of the equation.

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify both sides of the equation.

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

Cancel the common factor.

Rewrite the expression.

Simplify <math><mstyle displaystyle="true"><mn>2</mn><mo>⋅</mo><mrow><mo>(</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>135</mn><mo>)</mo></mrow></mstyle></math> .

Apply the distributive property.

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

Cancel the common factor.

Rewrite the expression.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>135</mn></mstyle></math> .

The basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn><mo>)</mo></mrow></mstyle></math> will occur at <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn><mo>,</mo><mn>3</mn><mi>π</mi><mo>+</mo><mn>270</mn><mo>)</mo></mrow></mstyle></math> , where <math><mstyle displaystyle="true"><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi><mo>+</mo><mn>270</mn></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.

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn><mo>)</mo></mrow></mstyle></math> occur at <math><mstyle displaystyle="true"><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn></mstyle></math> , <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi><mo>+</mo><mn>270</mn></mstyle></math> , and every <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn><mo>+</mo><mn>2</mn><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.

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn><mo>+</mo><mn>2</mn><mi>π</mi><mi>n</mi></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><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn><mo>+</mo><mn>2</mn><mi>π</mi><mi>n</mi></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Rewrite the expression as <math><mstyle displaystyle="true"><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>135</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mstyle></math> .

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"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> in the formula for period.

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</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>135</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfrac></mstyle></math>

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: <math><mstyle displaystyle="true"><mn>135</mn><mo>⋅</mo><mn>2</mn></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mn>135</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mn>270</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>270</mn></mstyle></math>

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

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: None

Period: <math><mstyle displaystyle="true"><mn>4</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>270</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>270</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</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><mo>-</mo><mi>π</mi><mo>+</mo><mn>270</mn><mo>+</mo><mn>2</mn><mi>π</mi><mi>n</mi></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Amplitude: None

Period: <math><mstyle displaystyle="true"><mn>4</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>270</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>270</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

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