Find Amplitude, Period, and Phase Shift y=3cos(0.5(x+pi))

$y = 3 \cos (0.5 (x + π))$

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

$a = 3$

$b = 0.5$

$c = - 1.57079632$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $3$

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 $0.5$ in the formula for period.

Period: $\frac{2 π}{| 0.5 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $0.5$ is $0.5$ .

Period: $\frac{2 π}{0.5}$

Replace $π$ with an approximation.

Period: $\frac{2 \cdot 3.14159265}{0.5}$

Multiply $2$ by $3.14159265$ .

Period: $\frac{6.2831853}{0.5}$

Divide $6.2831853$ by $0.5$ .

Period: $12.56637061$

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: $\frac{- 1.57079632}{0.5}$

Divide $- 1.57079632$ by $0.5$ .

Phase Shift: $- 3.14159265$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $12.56637061$

Phase Shift: $- 3.14159265$ ( $3.14159265$ to the left)

Vertical Shift: $0$

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