Find Amplitude, Period, and Phase Shift f(x)=3cos(1/2x)+2

$f (x) = 3 \cos (\frac{1}{2} x) + 2$

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.

$a = 3$

$b = \frac{1}{2}$

$c = 0$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

Find the period of $3 \cos (\frac{x}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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

$4 π$

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: $\frac{0}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 2$

Multiply $0$ by $2$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $4 π$

Phase Shift: None

Vertical Shift: $2$

Step 6

Find Amplitude, Period, and Phase Shift f(x)=3cos(1/2x)+2

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