Precalculus Examples

$f (x) = 2 \csc (4 x - π) + 1$

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.

$a = 2$

$b = 4$

$c = π$

$d = 1$

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

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Step 3.1

Find the period of $2 \csc (4 x - π)$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 3.1.4

Cancel the common factor of $2$ and $4$ .

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Step 3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.2

Find the period of $1$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 3.2.4

Cancel the common factor of $2$ and $4$ .

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Step 3.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.3

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

$\frac{π}{2}$

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: $\frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{2}$

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

Vertical Shift: $1$

Step 6