Precalculus Examples

$f (x) = 5 \tan (\frac{1}{4} x + \frac{π}{4}) - 2$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \tan (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = \tan (x)$ , $(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for $y = 5 \tan (\frac{x}{4} + \frac{π}{4}) - 2$ . Set the inside of the tangent function, $b x + c$ , for $y = a \tan (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = 5 \tan (\frac{x}{4} + \frac{π}{4}) - 2$ .

$\frac{x}{4} + \frac{π}{4} = - \frac{π}{2}$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.1.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{x}{4} = - \frac{π}{2} - \frac{π}{4}$

Step 1.2.1.2

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{4} = - \frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}$

Step 1.2.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$\frac{x}{4} = - \frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}$

Step 1.2.1.3.2

Multiply $2$ by $2$ .

$\frac{x}{4} = - \frac{π \cdot 2}{4} - \frac{π}{4}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{x}{4} = \frac{- π \cdot 2 - π}{4}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $2$ by $- 1$ .

$\frac{x}{4} = \frac{- 2 π - π}{4}$

Step 1.2.1.5.2

Subtract $π$ from $- 2 π$ .

$\frac{x}{4} = \frac{- 3 π}{4}$

Step 1.2.1.6

Move the negative in front of the fraction.

$\frac{x}{4} = - \frac{3 π}{4}$

Step 1.2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 3 π$

Step 1.3

Set the inside of the tangent function $\frac{x}{4} + \frac{π}{4}$ equal to $\frac{π}{2}$ .

$\frac{x}{4} + \frac{π}{4} = \frac{π}{2}$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.4.1.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{x}{4} = \frac{π}{2} - \frac{π}{4}$

Step 1.4.1.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{4} = \frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}$

Step 1.4.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$\frac{x}{4} = \frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}$

Step 1.4.1.3.2

Multiply $2$ by $2$ .

$\frac{x}{4} = \frac{π \cdot 2}{4} - \frac{π}{4}$

Step 1.4.1.4

Combine the numerators over the common denominator.

$\frac{x}{4} = \frac{π \cdot 2 - π}{4}$

Step 1.4.1.5

Simplify the numerator.

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Step 1.4.1.5.1

Move $2$ to the left of $π$ .

$\frac{x}{4} = \frac{2 \cdot π - π}{4}$

Step 1.4.1.5.2

Subtract $π$ from $2 π$ .

$\frac{x}{4} = \frac{π}{4}$

Step 1.4.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

Step 1.5

The basic period for $y = 5 \tan (\frac{x}{4} + \frac{π}{4}) - 2$ will occur at $(- 3 π, π)$ , where $- 3 π$ and $π$ are vertical asymptotes.

$(- 3 π, π)$

Step 1.6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 1.6.1

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{4}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 4$

Step 1.6.3

Move $4$ to the left of $π$ .

$4 π$

Step 1.7

The vertical asymptotes for $y = 5 \tan (\frac{x}{4} + \frac{π}{4}) - 2$ occur at $- 3 π$ , $π$ , and every $x = - 3 π + 4 π n$ , where $n$ is an integer.

$x = - 3 π + 4 π n$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - 3 π + 4 π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - 3 π + 4 π n$ where $n$ is an integer

Step 2

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

$a = 5$

$b = \frac{1}{4}$

$c = - \frac{π}{4}$

$d = - 2$

Step 3

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 4

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

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Step 4.1

Find the period of $5 \tan (\frac{x}{4} + \frac{π}{4})$ .

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Step 4.1.1

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

$\frac{π}{| b |}$

Step 4.1.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{π}{| \frac{1}{4} |}$

Step 4.1.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{4}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 4$

Step 4.1.5

Move $4$ to the left of $π$ .

$4 π$

Step 4.2

Find the period of $- 2$ .

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Step 4.2.1

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

$\frac{π}{| b |}$

Step 4.2.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{π}{| \frac{1}{4} |}$

Step 4.2.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{4}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 4$

Step 4.2.5

Move $4$ to the left of $π$ .

$4 π$

Step 4.3

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

$4 π$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- \frac{π}{4}}{\frac{1}{4}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{4} \cdot 4$

Step 5.4

Cancel the common factor of $4$ .

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Step 5.4.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

Phase Shift: $\frac{- π}{4} \cdot 4$

Step 5.4.2

Cancel the common factor.

Phase Shift: $\frac{- π}{4} \cdot 4$

Step 5.4.3

Rewrite the expression.

Phase Shift: $- π$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $4 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: $- 2$

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: $x = - 3 π + 4 π n$ where $n$ is an integer

Amplitude: None

Period: $4 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: $- 2$

Step 8