Find Amplitude, Period, and Phase Shift y=-6sin(3pix+4)-2

$y = - 6 \sin (3 π x + 4) - 2$

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

$a = - 6$

$b = 3 π$

$c = - 4$

$d = - 2$

Find the amplitude $| a |$ .

Amplitude: $6$

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

Find the period of $- 6 \sin (3 π x + 4)$ .

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 period of $- 2$ .

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.

$\frac{2}{3}$

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

$\frac{2}{3}$

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: $\frac{- 4}{3 π}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{4}{3 π}$

List the properties of the trigonometric function.

Amplitude: $6$

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

Phase Shift: $- \frac{4}{3 π}$ ( $\frac{4}{3 π}$ to the left)

Vertical Shift: $- 2$

Find Amplitude, Period, and Phase Shift y=-6sin(3pix+4)-2

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