Find Amplitude, Period, and Phase Shift y=-2cos(1/4x-pi/2)

$y = - 2 \cos (\frac{1}{4} x - \frac{π}{2})$

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

$a = - 2$

$b = \frac{1}{4}$

$c = \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $- 2 \cos (\frac{x}{4} - \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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

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

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

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

Phase Shift: $\frac{\frac{π}{2}}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot 4$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $π \cdot 2$

Move $2$ to the left of $π$ .

Phase Shift: $2 π$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $8 π$

Phase Shift: $2 π$ ( $2 π$ to the right)

Vertical Shift: None