# Precalculus Examples

For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the cosecant function, bx+c, for equal to to find where the vertical asymptote occurs for .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Divide by .

Set the inside of the cosecant function equal to .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

The basic period for will occur at , where and are vertical asymptotes.

Find the period 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 and is .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Since the graph of the function 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 .

Period:

Replace with in the formula for period.

Period:

Solve the equation.

The absolute value is the distance between a number and zero. The distance between and is .

Period:

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Period:

Divide by .

Period:

Period:

Period:

Period:

The phase shift of the function can be calculated from .

Phase Shift:

Replace the values of and in the equation for phase shift.

Phase Shift:

Divide by .

Phase Shift:

Phase Shift:

Find the vertical shift .

Vertical Shift:

List the properties of the trigonometric function.

Amplitude: None

Period:

Phase Shift: ( to the right)

Vertical Shift:

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

The exact value of is .

Multiply by .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

The exact value of is .

Multiply by .

Multiply by .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

Remove full rotations of until the angle is between and .

The exact value of is .

Multiply by .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

Remove full rotations of until the angle is between and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

The exact value of is .

Multiply by .

Multiply by .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

Remove full rotations of until the angle is between and .

The exact value of is .

Multiply by .

The final answer is .

List the points in a table.

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

Vertical Asymptotes: where is an integer

Amplitude: None

Period:

Phase Shift: ( to the right)

Vertical Shift: