Find Amplitude, Period, and Phase Shift f(x)=3sin(4x+2)-1

$f (x) = 3 \sin (4 x + 2) - 1$

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.

$a = 3$

$b = 4$

$c = - 2$

$d = - 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

Find the period of $3 \sin (4 x + 2)$ .

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.

$\frac{π}{2}$

Find the period of $- 1$ .

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.

$\frac{π}{2}$

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

$\frac{π}{2}$

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: $\frac{- 2}{4}$

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

Phase Shift: $\frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{- 1}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{π}{2}$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: $- 1$

Step 6

Find Amplitude, Period, and Phase Shift f(x)=3sin(4x+2)-1

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