Find Amplitude, Period, and Phase Shift f(t)=25sin(160pit)+115

$f (t) = 25 \sin (160 π t) + 115$

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

$a = 25$

$b = 160 π$

$c = 0$

$d = 115$

Find the amplitude $| a |$ .

Amplitude: $25$

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 $160 π$ in the formula for period.

Period: $\frac{2 π}{| 160 π |}$

Solve the equation.

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$160 π$ is approximately $502.65482457$ which is positive so remove the absolute value

Period: $\frac{2 π}{160 π}$

Cancel the common factor of $2$ and $160$ .

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Factor $2$ out of $2 π$ .

Period: $\frac{2 (π)}{160 π}$

Cancel the common factors.

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Factor $2$ out of $160 π$ .

Period: $\frac{2 (π)}{2 (80 π)}$

Cancel the common factor.

Period: $\frac{2 π}{2 (80 π)}$

Rewrite the expression.

Period: $\frac{π}{80 π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

Period: $\frac{π}{80 π}$

Rewrite the expression.

Period: $\frac{1}{80}$

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: $\frac{0}{160 π}$

Cancel the common factor of $0$ and $160$ .

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Factor $160$ out of $0$ .

Phase Shift: $\frac{160 (0)}{160 π}$

Cancel the common factors.

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Factor $160$ out of $160 π$ .

Phase Shift: $\frac{160 (0)}{160 (π)}$

Cancel the common factor.

Phase Shift: $\frac{160 \cdot 0}{160 π}$

Rewrite the expression.

Phase Shift: $\frac{0}{π}$

Divide $0$ by $π$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $115$

List the properties of the trigonometric function.

Amplitude: $25$

Period: $\frac{1}{80}$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $115$

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