Precalculus Examples

$f (x) = 2 \cot (x - \frac{π}{2}) - 4$

Step 1

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

$a = 2$

$b = 1$

$c = \frac{π}{2}$

$d = - 4$

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

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Step 3.1

Find the period of $2 \cot (x - \frac{π}{2})$ .

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Step 3.1.1

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

$\frac{π}{| b |}$

Step 3.1.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 3.1.4

Divide $π$ by $1$ .

$π$

Step 3.2

Find the period of $- 4$ .

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Step 3.2.1

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

$\frac{π}{| b |}$

Step 3.2.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 3.2.4

Divide $π$ by $1$ .

$π$

Step 3.3

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

$π$

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: $\frac{\frac{π}{2}}{1}$

Step 4.3

Divide $\frac{π}{2}$ by $1$ .

Phase Shift: $\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $- 4$

Step 6