Find Amplitude, Period, and Phase Shift f(x)=3cos(5x)+2

$f (x) = 3 \cos (5 x) + 2$

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.

$a = 3$

$b = 5$

$c = 0$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

Find the period of $3 \cos (5 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

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

$\frac{2 π}{5}$

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: $\frac{0}{5}$

Divide $0$ by $5$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{2 π}{5}$

Phase Shift: None

Vertical Shift: $2$

Step 6