Find Amplitude, Period, and Phase Shift f(x)=-4sin(2x+pi)-5

$f (x) = - 4 \sin (2 x + π) - 5$

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

$a = - 4$

$b = 2$

$c = - π$

$d = - 5$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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

Period: $\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

Period: $\frac{2 π}{| 2 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

Period: $\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Period: $\frac{2 π}{2}$

Divide $π$ by $1$ .

Period: $π$

Find the phase shift using the formula $\frac{c}{b}$ .

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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: $\frac{- π}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2}$

Find the vertical shift $d$ .

Vertical Shift: $- 5$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $π$

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

Vertical Shift: $- 5$

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