Find Amplitude, Period, and Phase Shift f(x)=1/9cos(x/5-(7pi)/4)-5

$f (x) = \frac{1}{9} \cos (\frac{x}{5} - \frac{7 π}{4}) - 5$

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

$a = \frac{1}{9}$

$b = \frac{1}{5}$

$c = \frac{7 π}{4}$

$d = - 5$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{9}$

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

Find the period of $\frac{\cos (\frac{x}{5} - \frac{7 π}{4})}{9}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{5}$ in the formula for period.

$\frac{2 π}{| \frac{1}{5} |}$

$\frac{1}{5}$ is approximately $0.2$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 5$

Multiply $5$ by $2$ .

$10 π$

Find the period of $- 5$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{5}$ in the formula for period.

$\frac{2 π}{| \frac{1}{5} |}$

$\frac{1}{5}$ is approximately $0.2$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 5$

Multiply $5$ by $2$ .

$10 π$

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

$10 π$

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: $\frac{\frac{7 π}{4}}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{7 π}{4} \cdot 5$

Multiply $\frac{7 π}{4} \cdot 5$ .

Combine $\frac{7 π}{4}$ and $5$ .

Phase Shift: $\frac{7 π \cdot 5}{4}$

Multiply $5$ by $7$ .

Phase Shift: $\frac{35 π}{4}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{9}$

Period: $10 π$

Phase Shift: $\frac{35 π}{4}$ ( $\frac{35 π}{4}$ to the right)

Vertical Shift: $- 5$

Find Amplitude, Period, and Phase Shift f(x)=1/9cos(x/5-(7pi)/4)-5

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