Precalculus Examples

$f (t) = \cot (9 t)$

Step 1

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

$a = 1$

$b = 9$

$c = 0$

$d = 0$

Step 2

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 3

Find the period of $\cot (9 t)$ .

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Step 3.1

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

$\frac{π}{| b |}$

Step 3.2

Replace $b$ with $9$ in the formula for period.

$\frac{π}{| 9 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$\frac{π}{9}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{9}$

Step 4.3

Divide $0$ by $9$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{9}$

Phase Shift: None

Vertical Shift: None

Step 6