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

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

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

$a = - \frac{1}{4}$

$b = \frac{π}{2}$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Rewrite the expression.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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: $\frac{0}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{π})$

Multiply $0$ by $\frac{2}{π}$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $4$

Phase Shift: None

Vertical Shift: None