Precalculus Examples

$G (x) = \csc (3 x - 3 π)$

Step 1

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

$a = 1$

$b = 3$

$c = 3 π$

$d = 0$

Step 2

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

Amplitude: None

Step 3

Find the period of $\csc (3 x - 3 π)$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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: $\frac{3 π}{3}$

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

Phase Shift: $\frac{3 π}{3}$

Step 4.3.2

Divide $π$ by $1$ .

Phase Shift: $π$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{2 π}{3}$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

Step 6