*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

Cotangent is the reciprocal of the tangent function. It is an odd function defined by the reciprocal identity cot (x) = 1 / tan (x). The cotangent graph can be sketched by first sketching the graph of y = tan (x) and then estimating the reciprocal of tan (x). It has the same period as its reciprocal, the tangent function.

A cotangent graph is a discontinuous graph that is not defined for theta values such that the value of sin (θ) is equal to zero. You can observe from the sin graph that the cotangent function has vertical asymptotes at the zero values of the sine graph.

## Cotangent Graph Properties

**Amplitude**

The amplitude of cotangent is not defined. Since relative values of x for which the cotangent is not specified, they are unbounded. The graph of cotangent functions goes on unending in the vertical direction. For y = α cot (βx - c) + d, the amplitude, sometimes called the stretching factor, is equal to |α|. It means that one should multiply all points of the vertical axis (y-coordinates) in the graph by the value of α.

**Period**

For y = cot (x), the period is equal to π. It means that one cycle of a cotangent graph occurs between 0 and π. The period of y = cot βx is equal to π/β. For y = α cot (βx - c) + d, the period is π / |β|.

**Domain**

Ordered pairs of the form (x, cot x) make up the cotangent function. Since cot (x) = cos (x) / sin (x), the domain of the cotangent function is the set of real numbers, except those values of s for which sin (x) = 0. Hence the domain contains all elements x such that x ≠ nπ, where n is an integer. For the domain of the graph y = α cot (βx), the domain is x ≠ π/|β|k. For the cotangent function of format y = α cot (βx - c) + d, the domain is x ≠ c/β + π/|β|k, where k is an integer.

**Range**

The range of the cotangent function is all real numbers. For the graph y = α cot (βx), it is (-∞, ∞) and for y = α cot (βx - c) + d, the range of the cotangent function with this arrangement is (-∞, -|α|] U [|α|, ∞).

**Continuity**

Each function is discontinuous at values of x, for which its reciprocal is zero. In the cotangent function’s case, it is not continuous at values of x, for which tangent is zero. The y = cot (x) graph is erratic at nπ, where n is an integer.

**Vertical Asymptotes**

Like the usual graph of inverse functions, the cotangent function has vertical asymptotes at the end of one cycle. The vertical asymptotes for the graph y = α cot (βx) occur at x = nπ / |β|, where n is an integer. On the other hand, for y = α cot (βx - c) + d, the vertical asymptotes occur at x = c/β + nπ/|β|, where n is an integer. Note that the vertical asymptotes of a cotangent function are where the denominator of the cot (x) has a value zero.

**X-intercepts**

The asymptotes of y = tan (x) are the x-intercepts of the graph y = cot (x). On the other hand, the x-intercepts of y = tan (x) are the asymptotes of y = cot (x).

**Vertical Shift**

The vertical shift of a cotangent function with equation y = α cot (βx - c) + d is equal to d moving upward. Otherwise, if the d is negative, then it shifts downward.

**Horizontal Shift**

The horizontal shift or the phase shift of a cotangent graph is equal to c/β, where positive results mean the graph shifts to the left and vice versa. For instance, given y = α cot (βx - c) + d, equate the terms inside the parenthesis to zero.

βx - c = 0

x = c/β

## How to Graph a Cotangent Function

As you can observe, the equation y = cot (x) graph is the opposite of the graph of its corresponding function, tangent. When doing the graphs of cotangent functions, do not get overwhelmed and try to compute and plot many points. You need to know where the graph is zero, where it is equal to 1, the location of the vertical asymptotes, and identify if there are horizontal and vertical shifts.

Note that there are various forms of cotangent functions. It can be of the form y = α cot (βx) or the form y = α cot (βx - c) + d with vertical and horizontal shifts. We can graph cotangent functions by following the step-by-step procedure shown below.

- Express the function in the simplest form f(x) = α cot (βx + c) + d.
- Determine the fundamental properties. Identify the parameters, including the stretching factor/amplitude, period, horizontal and vertical shifts, etc.
- Find the vertical asymptotes.
- Find the values for the domain and range.
- Determine the x-intercepts. It is good preparation for plotting reference points.
- Identify the vertical and horizontal shifts, if there are any.
- Evaluate and graph the cotangent function.

## Example 1: Graphing a Simple Cotangent Function

Determine the parameters of the cotangent equation y = 4cot (4x).

**Solution**

The given cotangent function is already in the y = α cot (βx). Next, identify the fundamental properties or parameters needed to graph the cotangent function like amplitude, period, etc.

**Amplitude: **The coefficient 4 is the amplitude of y = 4cot (4x).

Amplitude = |α|

Amplitude = |4|

**Period: **Solve for the period of y = 4cot (4x) using the formula P = π / |β|. We can identify that the value of β from the given equation is equal to 4.

P = π / |β|

P = π / |4|

P = π / 4

**Domain: **The domain of a cotangent function is the set of input or argument values for which it is defined and accurate.

πn / 4 < x < π / 4 + πn / 4

**Vertical Asymptotes: **Solve for the vertical asymptotes of the cotangent function using the general formula. The result shown below means that the one vertical asymptote appears at π / 4 and occurs every P = π / 4.

x = c/β + nπ/|β|

x = 0 + nπ / 4

x = nπ / 4

**X-intercepts: **Next, plot reference points of the graph by getting the x-intercepts. Substitute y = 0 to the given cotangent equation.

At y = 0,

y = 4 cot (4x)

0 = 4 cot (4(x))

x = π/8 + πn/4

Where n is any integer

At n =0, x = π/8

y = 4 cot (4x)

y = 4 cot (4(π/8))

y = 0

At n = 1, x = 3π/8

y = 4 cot (4x)

y = 4 cot (4(3π/8))

y = 0

At n = -1, x = -π/8

y = 4 cot (4x)

y = 4 cot (4(-π/8))

y = 0

You can observe that the value of y is zero for x with values π/8 + πn/4. Therefore, the axis interception points occur at (π/8 + πn/4, 0), where n is an integer.

Finally, sketch the graph of the given function given the period is π / 4, one of the vertical asymptotes is π / 4, y = 0 when x = π/8, x = -π/8, and x = 3π/8.

## Example 2: Cotangent Function with Horizontal and Vertical Shift

Sketch the cotangent graph of the equation y = 5cot (πx/8 - π/2) + 3.

**Solution**

The given cotangent equation has both phase shift and vertical shift. The first step is to identify the critical properties of the given cotangent equation to sketch its graph.

**Amplitude: **The coefficient 5 is the amplitude of y = 5cot (πx/8 - π/2) + 3. Therefore, the stretching factor is 5.

Amplitude = |α|

Amplitude = |5|

Amplitude = 5

**Period: **Solve for the period of y = 5 cot (πx/8 - π/2) + 3 using the formula P = π / |β|. We can identify that the value of β from the given equation is equal to π/8.

P = π / |β|

P = π / |π/8|

P = 8

**Vertical Asymptotes: **Solve for the vertical asymptotes of the cotangent function using the general formula.

x = c/β + nπ/|β|

x = (-π/2) / (π/8) + nπ / (π/8)

x = -4 + 8n

**Horizontal Shift: **The graph shifts four units to the right.

Phase shift = c/β

Phase shift = -π/2 / (π/8)

Phase shift = -4

**Vertical Shift: **The vertical shift of the cotangent equation y = 5cot (πx/8 - π/2) + 3 equals three units that shift upward.

**Inflection Points: **Find the inflection points to see where some of the graph's points are plotted in the cartesian coordinate system.

At y = 0

0 = 5 cot (πx/8 - π/2) + 3

x = 9.376 + 8n

x = 17.376 + 8n

At n = 1

x = 9.376 + 8(1)

x = 17.376

At n = -1

x = 9.376 + 8(-1)

x = 1.376

At n = -2

x = 9.376 + 8(-2)

x = -6.624

Finally, sketch the graph of the cotangent equation with both phase shift and vertical shift.

## Example 3: Sketching a Cotangent Function with Phase Shift

Draw the cotangent graph of the equation y = -3cot (x/2 + π/3).

**Solution**

The given cotangent function is already in the y = α cot (βx + c). Identify the parameters needed to graph the cotangent equation.

**Amplitude: **The coefficient -3 is the amplitude of y = -3 cot (x/2 + π/3).

Amplitude = |α|

Amplitude = |-3|

Amplitude = 3

**Period: **Solve for the period of y =-3 cot (x/2 + π/3) using the formula P = π / |β|. We can identify that the value of β from the given equation is equal to 1/2.

P = π / |β|

P = π / |1/2|

P = 2π

**Domain: **The domain of a cotangent function is the set of input or argument values for which the function is defined and real. The domain of y = -3 cot (x/2 + π/3) is shown below.

2πn ≤ x < 4π/3 + 2πn

**Vertical Asymptotes: **Solve for the vertical asymptotes of the cotangent function using the formula. From the result shown below, it means that the vertical asymptote appears at 2π/3 + 2πn.

x = c/β + nπ/|β|

x = (π/3) / (½) + nπ / (½)

x = 2π/3 + 2πn

**X-intercepts: **All inflection points are in a domain and not on domain edges.

At y =0

y = -3 cot (x/2 + π/3)

0 = -3 cot (x/2 + π/3)

x = π/3 + 2πn

At n = 1

x = π/3 + 2π(1)

x = 7π/3

At n = -1

x = π/3 + 2π(-1)

x = -5π/3

**Horizontal Shift/Phase Shift: **There is a horizontal shift in the cotangent graph since there is a constant added inside the grouping symbols of the function. Use the formula for horizontal shift c/β. Since the resulting value for phase shift is positive, then the graph shifts horizontally to the right.

Phase shift = -c/β

Phase shift = -π/3 / (½)

Phase shift = -2π/3

Or

βx + c = 0

x/2 + π/3 = 0

x = -2π/3

Finally, sketch the cotangent graph using the obtained parameters with a horizontal shift of 2π/3 to the right.

## Example 4: Graphing a Cotangent Function with a Stretch Factor

Graph the cotangent function y = 4 cot (⅕ x).

**Solution**

Describe the transformation of the cotangent function y = 4cot (⅕ x) and then graph it. Identify the parameters such as the stretch factor, period, domain, etc.

**Amplitude: **The coefficient 4 is the amplitude of y = 4cot (⅕ x). Therefore the given function has a stretch factor of 4.

Amplitude = |α|

Amplitude = |4|

Amplitude = 4

**Period: **Solve for the period of y = 4cot (⅕ x) using the formula P = π / |β|. We can identify that the value of β from the given equation is equal to ⅕.

P = π / |β|

P = π / |⅕|

P = 5π

**Domain: **The domain of a cotangent function is the set of input or argument values for which it is defined and accurate.

5πn < x < 5π + 5π

**Vertical Asymptotes: **Solve for the vertical asymptotes of the cotangent function using the formula. From the result shown below, it means that the vertical asymptotes appear at every x = 5πn.

x = c/β + nπ/|β|

x = 0 + nπ / |⅕|

x = 5πn

**X-intercepts: **Find the x-intercepts of the graph to plot other essential points of the cotangent graph.

At y = 0,

0 = 4 cot (⅕ x)

y = 4 cot (⅕ x)

x = 5π/2 + 5πn

At n = 1,

x = 5π/2 + 5π(1)

x = 15π/2

At n = -1

x = 5π/2 + 5π(-1)

x = - 5π/2

Lastly, sketch the cotangent graph applying the obtained parameters, especially the stretch factor.

## Example 5: Graphing a Cotangent Function with Phase Shift

Sketch the graph y = cot (5x - π/2).

**Solution**

Locate the essential parameters such as period, horizontal shift, and vertical asymptotes needed to graph the given cotangent function.

**Period: **Solve for the period of y = 4cot (⅕ x) using the formula P = π / |β|. We can identify that the value of β from the given equation is equal to ⅕.

P = π / |β|

P = π / |5||

P = π / 5

**Vertical Asymptotes: **Solve for the vertical asymptotes of the cotangent function using the formula. From the result shown below, it means that the vertical asymptotes appear at every x = 5πn.

x = c/β + nπ/|β|

x = -π/2 / 5 + nπ / |5|

x = -π/10 + nπ/5

**Horizontal Shift: **Equate the terms inside the parenthesis to zero and solve for the value of x, which represents the horizontal shift of the cotangent equation. With the positive result, the phase shifts to the left direction.

5x - π/2 = 0

x = π/2 / 5

x = π/10

Lastly, sketch the cotangent graph.

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*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**