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

$y = 3 + 4 \sin (x - π)$

Rewrite the expression as $4 \sin (x - π) + 3$ .

$4 \sin (x - π) + 3$

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

$a = 4$

$b = 1$

$c = π$

$d = 3$

Find the amplitude $| a |$ .

Amplitude: $4$

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

Find the period of $4 \sin (x - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Find the period of $3$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$2 π$

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: $\frac{π}{1}$

Divide $π$ by $1$ .

Phase Shift: $π$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $2 π$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: $3$