Find Amplitude, Period, and Phase Shift g(x)=1/2cos(4pix)

$g (x) = \frac{1}{2} \cos (4 π x)$

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.

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

$b = 4 π$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

Find the period of $\frac{\cos (4 π x)}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4 π$ in the formula for period.

$\frac{2 π}{| 4 π |}$

$4 π$ is approximately $12.56637061$ which is positive so remove the absolute value

$\frac{2 π}{4 π}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4 π}$

Cancel the common factors.

Factor $2$ out of $4 π$ .

$\frac{2 (π)}{2 (2 π)}$

Cancel the common factor.

$\frac{2 π}{2 (2 π)}$

Rewrite the expression.

$\frac{π}{2 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π}{2 π}$

Rewrite the expression.

$\frac{1}{2}$

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: $\frac{0}{4 π}$

Cancel the common factor of $0$ and $4$ .

Factor $4$ out of $0$ .

Phase Shift: $\frac{4 (0)}{4 π}$

Cancel the common factors.

Factor $4$ out of $4 π$ .

Phase Shift: $\frac{4 (0)}{4 (π)}$

Cancel the common factor.

Phase Shift: $\frac{4 \cdot 0}{4 π}$

Rewrite the expression.

Phase Shift: $\frac{0}{π}$

Divide $0$ by $π$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

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

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

Phase Shift: None

Vertical Shift: None

Step 6

Find Amplitude, Period, and Phase Shift g(x)=1/2cos(4pix)

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