Find Amplitude, Period, and Phase Shift y=5sin((2x)/3-pi)

$y = 5 \sin (\frac{2 x}{3} - π)$

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

$a = 5$

$b = \frac{2}{3}$

$c = π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $5$

Find the period using the formula $\frac{2 π}{| b |}$ .

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

Period: $\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{3}$ in the formula for period.

Period: $\frac{2 π}{| \frac{2}{3} |}$

Solve the equation.

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$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

Period: $\frac{2 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Period: $2 π (\frac{3}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

Period: $2 (π) (\frac{3}{2})$

Cancel the common factor.

Period: $2 π (\frac{3}{2})$

Rewrite the expression.

Period: $π \cdot 3$

Move $3$ to the left of $π$ .

Period: $3 π$

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: $\frac{π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{3}{2})$

Combine $π$ and $\frac{3}{2}$ .

Phase Shift: $\frac{π \cdot 3}{2}$

Move $3$ to the left of $π$ .

Phase Shift: $\frac{3 π}{2}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $3 π$

Phase Shift: $\frac{3 π}{2}$ ( $\frac{3 π}{2}$ to the right)

Vertical Shift: $0$

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