Precalculus Examples

$y = - 3 \cos (- 2 x + \frac{π}{2})$

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.

$a = - 3$

$b = - 2$

$c = - \frac{π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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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 $- 2$ in the formula for period.

$\frac{2 π}{| - 2 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$\frac{2 π}{2}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.4.2

Divide $π$ by $1$ .

$π$

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: $\frac{- \frac{π}{2}}{- 2}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 4.5

Multiply $- \frac{π}{2} (- \frac{1}{2})$ .

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

Phase Shift: $1 (\frac{π}{2}) \cdot \frac{1}{2}$

Step 4.5.2

Multiply $\frac{π}{2}$ by $1$ .

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

Step 4.5.3

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

Phase Shift: $\frac{π}{2 \cdot 2}$

Step 4.5.4

Multiply $2$ by $2$ .

Phase Shift: $\frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: None

Step 6