Find Amplitude, Period, and Phase Shift

$y = \cos (3 x + \frac{π}{2})$

Step 1

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = 3$

$c = - \frac{π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\cos (3 x + \frac{π}{2})$ .

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Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- \frac{π}{2}}{3}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot \frac{1}{3}$

Step 4.4

Multiply $- \frac{π}{2} \cdot \frac{1}{3}$ .

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Step 4.4.1

Multiply $\frac{1}{3}$ by $\frac{π}{2}$ .

Phase Shift: $- \frac{π}{3 \cdot 2}$

Step 4.4.2

Multiply $3$ by $2$ .

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: None

Step 6

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