Trigonometry Examples

$y = - 2 + \frac{2}{3} \cdot \tan (\frac{3}{4} x - π)$

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 $y = - 2 + \frac{2 \tan (\frac{3 x}{4} - π)}{3}$ . 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 $y = - 2 + \frac{2 \tan (\frac{3 x}{4} - π)}{3}$ .

$\frac{3 x}{4} - π = - \frac{π}{2}$

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

Add $π$ to both sides of the equation.

$\frac{3 x}{4} = - \frac{π}{2} + π$

Step 1.2.1.2

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

$\frac{3 x}{4} = - \frac{π}{2} + π \cdot \frac{2}{2}$

Step 1.2.1.3

Combine $π$ and $\frac{2}{2}$ .

$\frac{3 x}{4} = - \frac{π}{2} + \frac{π \cdot 2}{2}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} = \frac{- π + π \cdot 2}{2}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Move $2$ to the left of $π$ .

$\frac{3 x}{4} = \frac{- π + 2 \cdot π}{2}$

Step 1.2.1.5.2

Add $- π$ and $2 π$ .

$\frac{3 x}{4} = \frac{π}{2}$

Step 1.2.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 1.2.3.1.1.1

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} \cdot \frac{π}{2}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify $\frac{4}{3} \cdot \frac{π}{2}$ .

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Step 1.2.3.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{3} \cdot \frac{π}{2}$

Step 1.2.3.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{3} \cdot \frac{π}{2}$

Step 1.2.3.2.1.1.3

Rewrite the expression.

$x = \frac{2}{3} π$

Step 1.2.3.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$x = \frac{2 π}{3}$

Step 1.3

Set the inside of the tangent function $\frac{3 x}{4} - π$ equal to $\frac{π}{2}$ .

$\frac{3 x}{4} - π = \frac{π}{2}$

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

Add $π$ to both sides of the equation.

$\frac{3 x}{4} = \frac{π}{2} + π$

Step 1.4.1.2

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

$\frac{3 x}{4} = \frac{π}{2} + π \cdot \frac{2}{2}$

Step 1.4.1.3

Combine $π$ and $\frac{2}{2}$ .

$\frac{3 x}{4} = \frac{π}{2} + \frac{π \cdot 2}{2}$

Step 1.4.1.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} = \frac{π + π \cdot 2}{2}$

Step 1.4.1.5

Simplify the numerator.

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Step 1.4.1.5.1

Move $2$ to the left of $π$ .

$\frac{3 x}{4} = \frac{π + 2 \cdot π}{2}$

Step 1.4.1.5.2

Add $π$ and $2 π$ .

$\frac{3 x}{4} = \frac{3 π}{2}$

Step 1.4.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3

Simplify both sides of the equation.

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Step 1.4.3.1

Simplify the left side.

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Step 1.4.3.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 1.4.3.1.1.1

Cancel the common factor of $4$ .

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Step 1.4.3.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.1.1.2

Cancel the common factor of $3$ .

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Step 1.4.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.2

Simplify the right side.

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Step 1.4.3.2.1

Simplify $\frac{4}{3} \cdot \frac{3 π}{2}$ .

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Step 1.4.3.2.1.1

Cancel the common factor of $2$ .

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Step 1.4.3.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.2.1.1.3

Rewrite the expression.

$x = \frac{2}{3} (3 π)$

Step 1.4.3.2.1.2

Cancel the common factor of $3$ .

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Step 1.4.3.2.1.2.1

Factor $3$ out of $3 π$ .

$x = \frac{2}{3} (3 (π))$

Step 1.4.3.2.1.2.2

Cancel the common factor.

$x = \frac{2}{3} (3 π)$

Step 1.4.3.2.1.2.3

Rewrite the expression.

$x = 2 π$

Step 1.5

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

$(\frac{2 π}{3}, 2 π)$

Step 1.6

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

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Step 1.6.1

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{4}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{3}$

Step 1.6.3

Combine $π$ and $\frac{4}{3}$ .

$\frac{π \cdot 4}{3}$

Step 1.6.4

Move $4$ to the left of $π$ .

$\frac{4 π}{3}$

Step 1.7

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

$x = \frac{2 π}{3} + \frac{4 π n}{3}$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{2 π}{3} + \frac{4 π n}{3}$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{2 π}{3} + \frac{4 π n}{3}$ where $n$ is an integer

Step 2

Rewrite the expression as $\frac{2 \tan (\frac{3 x}{4} - π)}{3} - 2$ .

$\frac{2 \tan (\frac{3 x}{4} - π)}{3} - 2$

Step 3

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

$a = \frac{2}{3}$

$b = \frac{3}{4}$

$c = π$

$d = - 2$

Step 4

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 5

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

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Step 5.1

Find the period of $\frac{2 \tan (\frac{3 x}{4} - π)}{3}$ .

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Step 5.1.1

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

$\frac{π}{| b |}$

Step 5.1.2

Replace $b$ with $\frac{3}{4}$ in the formula for period.

$\frac{π}{| \frac{3}{4} |}$

Step 5.1.3

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{4}}$

Step 5.1.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{3}$

Step 5.1.5

Combine $π$ and $\frac{4}{3}$ .

$\frac{π \cdot 4}{3}$

Step 5.1.6

Move $4$ to the left of $π$ .

$\frac{4 π}{3}$

Step 5.2

Find the period of $- 2$ .

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Step 5.2.1

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

$\frac{π}{| b |}$

Step 5.2.2

Replace $b$ with $\frac{3}{4}$ in the formula for period.

$\frac{π}{| \frac{3}{4} |}$

Step 5.2.3

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{4}}$

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{3}$

Step 5.2.5

Combine $π$ and $\frac{4}{3}$ .

$\frac{π \cdot 4}{3}$

Step 5.2.6

Move $4$ to the left of $π$ .

$\frac{4 π}{3}$

Step 5.3

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

$\frac{4 π}{3}$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

Phase Shift: $\frac{π}{\frac{3}{4}}$

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{4}{3})$

Step 6.4

Combine $π$ and $\frac{4}{3}$ .

Phase Shift: $\frac{π \cdot 4}{3}$

Step 6.5

Move $4$ to the left of $π$ .

Phase Shift: $\frac{4 π}{3}$

Step 7

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{4 π}{3}$ ( $\frac{4 π}{3}$ to the right)

Vertical Shift: $- 2$

Step 8

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

Vertical Asymptotes: $x = \frac{2 π}{3} + \frac{4 π n}{3}$ where $n$ is an integer

Amplitude: None

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{4 π}{3}$ ( $\frac{4 π}{3}$ to the right)

Vertical Shift: $- 2$

Step 9