Find Amplitude, Period, and Phase Shift y=6sin(8(x-pi/2))

$y = 6 \sin (8 (x - \frac{π}{2}))$

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

$a = 6$

$b = 8$

$c = 4 π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $6$

Find the period of $6 \sin (8 x - 4 π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $8$ in the formula for period.

$\frac{2 π}{| 8 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{2 π}{8}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 π}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 4}$

Rewrite the expression.

$\frac{π}{4}$

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: $\frac{4 π}{8}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 π$ .

Phase Shift: $\frac{4 (π)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

Phase Shift: $\frac{4 π}{4 \cdot 2}$

Cancel the common factor.

Phase Shift: $\frac{4 π}{4 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $6$

Period: $\frac{π}{4}$

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

Vertical Shift: None