Find Amplitude, Period, and Phase Shift y=1/4sin(4(x+pi/6))

$y = \frac{1}{4} \sin (4 (x + \frac{π}{6}))$

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

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

$b = 4$

$c = - \frac{2 π}{3}$

$d = 0$

Find the amplitude $| a |$ .

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

Find the period of $\frac{\sin (4 x + \frac{2 π}{3})}{4}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{- \frac{2 π}{3}}{4}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{2 π}{3} \cdot \frac{1}{4}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2 π}{3}$ into the numerator.

Phase Shift: $\frac{- 2 π}{3} \cdot \frac{1}{4}$

Factor $2$ out of $- 2 π$ .

Phase Shift: $\frac{2 (- π)}{3} \cdot \frac{1}{4}$

Factor $2$ out of $4$ .

Phase Shift: $\frac{2 (- π)}{3} \cdot \frac{1}{2 (2)}$

Cancel the common factor.

Phase Shift: $\frac{2 (- π)}{3} \cdot \frac{1}{2 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{- π}{3} \cdot \frac{1}{2}$

Multiply $\frac{- π}{3}$ by $\frac{1}{2}$ .

Phase Shift: $\frac{- π}{3 \cdot 2}$

Simplify the expression.

Multiply $3$ by $2$ .

Phase Shift: $\frac{- π}{6}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{6}$

List the properties of the trigonometric function.

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

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: None

Find Amplitude, Period, and Phase Shift y=1/4sin(4(x+pi/6))

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