Graph y=cot(4x-pi/2)-3

$y = \cot (4 x - \frac{π}{2}) - 3$

Find the asymptotes.

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

$4 x - \frac{π}{2} = 0$

Solve for $x$ .

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Add $\frac{π}{2}$ to both sides of the equation.

$4 x = \frac{π}{2}$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = \frac{π}{2}$ by $4$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{π}{2} \cdot \frac{1}{4}$

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

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Multiply $\frac{π}{2}$ and $\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{π}{8}$

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

$4 x - \frac{π}{2} = π$

Solve for $x$ .

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

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Add $\frac{π}{2}$ to both sides of the equation.

$4 x = π + \frac{π}{2}$

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$4 x = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$4 x = \frac{3 π}{2}$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = \frac{3 π}{2}$ by $4$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{3 π}{2} \cdot \frac{1}{4}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{4}$ .

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Multiply $\frac{3 π}{2}$ and $\frac{1}{4}$ .

$x = \frac{3 π}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{3 π}{8}$

The basic period for $y = \cot (4 x - \frac{π}{2}) - 3$ will occur at $(\frac{π}{8}, \frac{3 π}{8})$ , where $\frac{π}{8}$ and $\frac{3 π}{8}$ are vertical asymptotes.

$(\frac{π}{8}, \frac{3 π}{8})$

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

$\frac{π}{4}$

The vertical asymptotes for $y = \cot (4 x - \frac{π}{2}) - 3$ occur at $\frac{π}{8}$ , $\frac{3 π}{8}$ , and every $x = \frac{π}{8} + \frac{π n}{4}$ , where $n$ is an integer.

$x = \frac{π}{8} + \frac{π n}{4}$

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{8} + \frac{π n}{4}$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{8} + \frac{π n}{4}$ where $n$ is an integer

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

$a = 1$

$b = 4$

$c = \frac{π}{2}$

$d = - 3$

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

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 4 |}$

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

$\frac{π}{4}$

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

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

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

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

Phase Shift: $\frac{\frac{π}{2}}{4}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{4}$

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

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Multiply $\frac{π}{2}$ and $\frac{1}{4}$ .

Phase Shift: $\frac{π}{2 \cdot 4}$

Multiply $2$ by $4$ .

Phase Shift: $\frac{π}{8}$

Find the vertical shift $d$ .

Vertical Shift: $- 3$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{4}$

Phase Shift: $\frac{π}{8}$ ( $\frac{π}{8}$ to the right)

Vertical Shift: $- 3$

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

Vertical Asymptotes: $x = \frac{π}{8} + \frac{π n}{4}$ where $n$ is an integer

Amplitude: None

Period: $\frac{π}{4}$

Phase Shift: $\frac{π}{8}$ ( $\frac{π}{8}$ to the right)

Vertical Shift: $- 3$

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