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

Find the amplitude .

Amplitude:

The period of the function can be calculated using .

Replace with in the formula for period.

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

Divide by .

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:

Period:

Phase Shift: ( to the left)

Vertical Shift:

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Add and .

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

To write as a fraction with a common denominator, multiply by .

Combine fractions.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Add and .

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

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.

To write as a fraction with a common denominator, multiply by .

Combine fractions.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Add and .

Subtract full rotations of until the angle is greater than or equal to and less than .

The exact value of is .

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.

Amplitude:

Period:

Phase Shift: ( to the left)

Vertical Shift:

Graph y=cos(x+pi)