Algebra Examples

$f (x) = \cot (\frac{1}{2} x - π)$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (\frac{x}{2} - π)$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (\frac{x}{2} - π)$ .

$\frac{x}{2} - π = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $π$ to both sides of the equation.

$\frac{x}{2} = π$

Step 1.2.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 π$

Step 1.2.3

Simplify the left side.

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Step 1.2.3.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 π$

Step 1.2.3.1.2

Rewrite the expression.

$x = 2 π$

Step 1.3

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

$\frac{x}{2} - π = π$

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.

$\frac{x}{2} = π + π$

Step 1.4.1.2

Add $π$ and $π$ .

$\frac{x}{2} = 2 π$

Step 1.4.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π)$

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

Cancel the common factor of $2$ .

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Step 1.4.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (2 π)$

Step 1.4.3.1.1.2

Rewrite the expression.

$x = 2 (2 π)$

Step 1.4.3.2

Simplify the right side.

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Step 1.4.3.2.1

Multiply $2$ by $2$ .

$x = 4 π$

Step 1.5

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

$(2 π, 4 π)$

Step 1.6

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

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Step 1.6.1

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 1.6.3

Move $2$ to the left of $π$ .

$2 π$

Step 1.7

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

$x = 2 π + 2 π n$

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π + 2 π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π + 2 π n$ where $n$ is an integer

Step 2

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

$a = 1$

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

$c = π$

$d = 0$

Step 3

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

Amplitude: None

Step 4

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

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Step 4.1

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

$\frac{π}{| b |}$

Step 4.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 4.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 4.5

Move $2$ to the left of $π$ .

$2 π$

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: $\frac{π}{\frac{1}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π \cdot 2$

Step 5.4

Move $2$ to the left of $π$ .

Phase Shift: $2 π$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $2 π$

Phase Shift: $2 π$ ( $2 π$ to the right)

Vertical Shift: None

Step 7

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

Vertical Asymptotes: $x = 2 π + 2 π n$ where $n$ is an integer

Amplitude: None

Period: $2 π$

Phase Shift: $2 π$ ( $2 π$ to the right)

Vertical Shift: None

Step 8