Precalculus Examples

$f (x) = 7 \csc (\frac{1}{3} \cdot (π x) + \frac{1}{6} \cdot π)$

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.

$a = 7$

$b = \frac{π}{3}$

$c = - \frac{π}{6}$

$d = 0$

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 $7 \csc (\frac{π x}{3} + \frac{π}{6})$ .

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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 $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 3.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 3.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 3.5.3

Rewrite the expression.

$2 \cdot 3$

Step 3.6

Multiply $2$ by $3$ .

$6$

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: $\frac{- \frac{π}{6}}{\frac{π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot \frac{3}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot \frac{3}{π}$

Step 4.4.2

Factor $π$ out of $- π$ .

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{3}{π}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{3}{π}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- 1}{6} \cdot 3$

Step 4.5

Cancel the common factor of $3$ .

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Step 4.5.1

Factor $3$ out of $6$ .

Phase Shift: $\frac{- 1}{3 (2)} \cdot 3$

Step 4.5.2

Cancel the common factor.

Phase Shift: $\frac{- 1}{3 \cdot 2} \cdot 3$

Step 4.5.3

Rewrite the expression.

Phase Shift: $\frac{- 1}{2}$

Step 4.6

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $6$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: None

Step 6