Find Amplitude, Period, and Phase Shift y=4picos(0.2(x+pi))

$y = 4 π \cos (0.2 (x + π))$

Step 1

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

$a = 4 π$

$b = 0.2$

$c = - 0.62831853$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4 π$

Step 3

Find the period of $4 π \cos (0.2 x + 0.62831853)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $0.2$ in the formula for period.

$\frac{2 π}{| 0.2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $0.2$ is $0.2$ .

$\frac{2 π}{0.2}$

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.2}$

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.2}$

Divide $6.2831853$ by $0.2$ .

$31.41592653$

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: $\frac{- 0.62831853}{0.2}$

Divide $- 0.62831853$ by $0.2$ .

Phase Shift: $- 3.14159265$

Step 5

List the properties of the trigonometric function.

Amplitude: $4 π$

Period: $31.41592653$

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

Vertical Shift: None

Step 6