Find Amplitude, Period, and Phase Shift f(x)=5sin((2pi)/11(x-8))+12

$f (x) = 5 \sin (\frac{2 π}{11} (x - 8)) + 12$

Step 1

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

$a = 5$

$b = \frac{2 π}{11}$

$c = \frac{16 π}{11}$

$d = 12$

Step 2

Find the amplitude $| a |$ .

Amplitude: $5$

Step 3

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

Find the period of $5 \sin (\frac{2 π x}{11} - \frac{16 π}{11})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{11}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{11} |}$

$\frac{2 π}{11}$ is approximately $0.57119866$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{11}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{11}{2 π}$

Cancel the common factor of $2 π$ .

Cancel the common factor.

$2 π \frac{11}{2 π}$

Rewrite the expression.

$11$

Find the period of $12$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{11}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{11} |}$

$\frac{2 π}{11}$ is approximately $0.57119866$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{11}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{11}{2 π}$

Cancel the common factor of $2 π$ .

Cancel the common factor.

$2 π \frac{11}{2 π}$

Rewrite the expression.

$11$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$11$

Step 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: $\frac{\frac{16 π}{11}}{\frac{2 π}{11}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{16 π}{11} \cdot \frac{11}{2 π}$

Cancel the common factor of $2 π$ .

Factor $2 π$ out of $16 π$ .

Phase Shift: $\frac{2 π (8)}{11} \cdot \frac{11}{2 π}$

Cancel the common factor.

Phase Shift: $\frac{2 π \cdot 8}{11} \cdot \frac{11}{2 π}$

Rewrite the expression.

Phase Shift: $\frac{8}{11} \cdot 11$

Cancel the common factor of $11$ .

Cancel the common factor.

Phase Shift: $\frac{8}{11} \cdot 11$

Rewrite the expression.

Phase Shift: $8$

Step 5

List the properties of the trigonometric function.

Amplitude: $5$

Period: $11$

Phase Shift: $8$ ( $8$ to the right)

Vertical Shift: $12$

Step 6

Find Amplitude, Period, and Phase Shift f(x)=5sin((2pi)/11(x-8))+12

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