Precalculus Examples

$f (x) = \frac{1}{2} \cdot \tan (π x - π)$

Step 1

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

$a = \frac{1}{2}$

$b = π$

$c = π$

$d = 0$

Step 2

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

Amplitude: None

Step 3

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

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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 $π$ in the formula for period.

$\frac{π}{| π |}$

Step 3.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{π}{π}$

Step 3.4

Cancel the common factor of $π$ .

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Step 3.4.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.4.2

Rewrite the expression.

$1$

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: $\frac{π}{π}$

Step 4.3

Cancel the common factor of $π$ .

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Step 4.3.1

Cancel the common factor.

Phase Shift: $\frac{π}{π}$

Step 4.3.2

Rewrite the expression.

Phase Shift: $1$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $1$

Phase Shift: $1$ ( $1$ to the right)

Vertical Shift: None

Step 6