Find Amplitude, Period, and Phase Shift y=-3/2cos((3x)/4)

$y = - \frac{3}{2} \cos (\frac{3 x}{4})$

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

$a = - \frac{3}{2}$

$b = \frac{3}{4}$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{3}{2}$

Find the period of $- \frac{3 \cos (\frac{3 x}{4})}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3}{4}$ in the formula for period.

$\frac{2 π}{| \frac{3}{4} |}$

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{3}$

Multiply $2 π \frac{4}{3}$ .

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Combine $\frac{4}{3}$ and $2$ .

$\frac{4 \cdot 2}{3} π$

Multiply $4$ by $2$ .

$\frac{8}{3} π$

Combine $\frac{8}{3}$ and $π$ .

$\frac{8 π}{3}$

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

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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: $\frac{0}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{4}{3})$

Multiply $0$ by $\frac{4}{3}$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{3}{2}$

Period: $\frac{8 π}{3}$

Phase Shift: None

Vertical Shift: None

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