Find Amplitude, Period, and Phase Shift f(x)=-4sin(3x-pi)+5

$f (x) = - 4 \sin (3 x - π) + 5$

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.

$a = - 4$

$b = 3$

$c = π$

$d = 5$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

Find the period of $- 4 \sin (3 x - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Find the period of $5$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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

$\frac{2 π}{3}$

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: $\frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{3}$ ( $\frac{π}{3}$ to the right)

Vertical Shift: $5$

Step 6