Find Amplitude, Period, and Phase Shift y=-6sin(4x+pi/2)

$y = - 6 \sin (4 x + \frac{π}{2})$

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.

$a = - 6$

$b = 4$

$c = - \frac{π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $6$

Step 3

Find the period of $- 6 \sin (4 x + \frac{π}{2})$ .

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.

$\frac{π}{2}$

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: $\frac{- \frac{π}{2}}{4}$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $- \frac{π}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{4}$ by $\frac{π}{2}$ .

Phase Shift: $- \frac{π}{4 \cdot 2}$

Multiply $4$ by $2$ .

Phase Shift: $- \frac{π}{8}$

Step 5

List the properties of the trigonometric function.

Amplitude: $6$

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{8}$ ( $\frac{π}{8}$ to the left)

Vertical Shift: None

Step 6

Find Amplitude, Period, and Phase Shift y=-6sin(4x+pi/2)

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