Find Amplitude, Period, and Phase Shift m(t)=7cos(t+3)

$m (t) = 7 \cos (t + 3)$

Step 1

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

$a = 7$

$b = 1$

$c = - 3$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $7$

Step 3

Find the period of $7 \cos (t + 3)$ .

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 $1$ .

$2 π$

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: $\frac{- 3}{1}$

Divide $- 3$ by $1$ .

Phase Shift: $- 3$

Step 5

List the properties of the trigonometric function.

Amplitude: $7$

Period: $2 π$

Phase Shift: $- 3$ ( $3$ to the left)

Vertical Shift: None

Step 6