Find Amplitude, Period, and Phase Shift y=5cos(3/5x-pi)+4

$y = 5 \cos (\frac{3}{5} x - π) + 4$

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.

$a = 5$

$b = \frac{3}{5}$

$c = π$

$d = 4$

Step 2

Find the amplitude $| a |$ .

Amplitude: $5$

Step 3

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

Find the period of $5 \cos (\frac{3 x}{5} - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3}{5}$ in the formula for period.

$\frac{2 π}{| \frac{3}{5} |}$

$\frac{3}{5}$ is approximately $0.6$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{3}$

Multiply $2 π \frac{5}{3}$ .

Combine $\frac{5}{3}$ and $2$ .

$\frac{5 \cdot 2}{3} π$

Multiply $5$ by $2$ .

$\frac{10}{3} π$

Combine $\frac{10}{3}$ and $π$ .

$\frac{10 π}{3}$

Find the period of $4$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3}{5}$ in the formula for period.

$\frac{2 π}{| \frac{3}{5} |}$

$\frac{3}{5}$ is approximately $0.6$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{3}$

Multiply $2 π \frac{5}{3}$ .

Combine $\frac{5}{3}$ and $2$ .

$\frac{5 \cdot 2}{3} π$

Multiply $5$ by $2$ .

$\frac{10}{3} π$

Combine $\frac{10}{3}$ and $π$ .

$\frac{10 π}{3}$

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

$\frac{10 π}{3}$

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: $\frac{π}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{5}{3})$

Combine $π$ and $\frac{5}{3}$ .

Phase Shift: $\frac{π \cdot 5}{3}$

Move $5$ to the left of $π$ .

Phase Shift: $\frac{5 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $5$

Period: $\frac{10 π}{3}$

Phase Shift: $\frac{5 π}{3}$ ( $\frac{5 π}{3}$ to the right)

Vertical Shift: $4$

Step 6

Find Amplitude, Period, and Phase Shift y=5cos(3/5x-pi)+4

Related Questions

Support LCMGCF.com by disabling your adblocker.