Find Amplitude, Period, and Phase Shift f(x)=3sin(3pix-1)

$f (x) = 3 \sin (3 π x - 1)$

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

$a = 3$

$b = 3 π$

$c = 1$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $3$

Find the period of $3 \sin (3 π x - 1)$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 3 π |}$

$3 π$ is approximately $9.42477796$ which is positive so remove the absolute value

$\frac{2 π}{3 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{2 π}{3 π}$

Rewrite the expression.

$\frac{2}{3}$

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: $\frac{1}{3 π}$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{2}{3}$

Phase Shift: $\frac{1}{3 π}$ ( $\frac{1}{3 π}$ to the right)

Vertical Shift: None