8.2 Graphs of the other trigonometric functions (Page 7/9)

Page 7 / 9

The cotangent graph has vertical asymptotes at each value of $x$ where $\tan x = 0;$ we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, $\cot x$ has vertical asymptotes at all values of $x$ where $\tan x = 0,$ and $\cot x = 0$ at all values of $x$ where $\tan x$ has its vertical asymptotes.

A graph of cotangent of x, with vertical asymptotes at multiples of pi. — The cotangent function

a general note label

Features of the graph of y = A Cot( Bx )

The stretching factor is $| A | .$
The period is $P = \frac{π}{| B |} .$
The domain is $x \neq \frac{π}{| B |} k,$ where $k$ is an integer.
The range is $(- \infty, \infty) .$
The asymptotes occur at $x = \frac{π}{| B |} k,$ where $k$ is an integer.
$y = A \cot (B x)$ is an odd function.

Graphing variations of y = cot x

We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.

y = A \cot (B x - C) + D

a general note label

Properties of the graph of y = A Cot( Bx −c)+ D

The stretching factor is $| A | .$
The period is $\frac{π}{| B |} .$
The domain is $x \neq \frac{C}{B} + \frac{π}{| B |} k,$ where $k$ is an integer.
The range is $(−∞, \infty) .$
The vertical asymptotes occur at $x = \frac{C}{B} + \frac{π}{| B |} k,$ where $k$ is an integer.
There is no amplitude.
$y = A \cot (B x)$ is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)

how to feature

Given a modified cotangent function of the form $f (x) = A \cot (B x),$ graph one period.

Express the function in the form $f (x) = A \cot (B x) .$
Identify the stretching factor, $| A | .$
Identify the period, $P = \frac{π}{| B |} .$
Draw the graph of $y = A \tan (B x) .$
Plot any two reference points.
Use the reciprocal relationship between tangent and cotangent to draw the graph of $y = A \cot (B x) .$
Sketch the asymptotes.

Graphing variations of the cotangent function

Determine the stretching factor, period, and phase shift of $y = 3 \cot (4 x),$ and then sketch a graph.

Step 1. Expressing the function in the form $f (x) = A \cot (B x)$ gives $f (x) = 3 \cot (4 x) .$
Step 2. The stretching factor is $| A | = 3.$
Step 3. The period is $P = \frac{π}{4} .$
Step 4. Sketch the graph of $y = 3 \tan (4 x) .$
Step 5. Plot two reference points. Two such points are $(\frac{π}{16}, 3)$ and $(\frac{3 π}{16}, −3) .$
Step 6. Use the reciprocal relationship to draw $y = 3 \cot (4 x) .$
Step 7. Sketch the asymptotes, $x = 0, x = \frac{π}{4} .$

The orange graph in [link] shows $y = 3 \tan (4 x)$ and the blue graph shows $y = 3 \cot (4 x) .$

A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi/4 and pi/4.

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how to feature

Given a modified cotangent function of the form $f (x) = A \cot (B x - C) + D,$ graph one period.

Express the function in the form $f (x) = A \cot (B x - C) + D .$
Identify the stretching factor, $| A | .$
Identify the period, $P = \frac{π}{| B |} .$
Identify the phase shift, $\frac{C}{B} .$
Draw the graph of $y = A \tan (B x)$ shifted to the right by $\frac{C}{B}$ and up by $D .$
Sketch the asymptotes $x = \frac{C}{B} + \frac{π}{| B |} k,$ where $k$ is an integer.
Plot any three reference points and draw the graph through these points.

Graphing a modified cotangent

Sketch a graph of one period of the function $f (x) = 4 \cot (\frac{π}{8} x - \frac{π}{2}) - 2.$

Step 1. The function is already written in the general form $f (x) = A \cot (B x - C) + D .$
Step 2. $A = 4,$ so the stretching factor is 4.
Step 3. $B = \frac{π}{8},$ so the period is $P = \frac{π}{| B |} = \frac{π}{\frac{π}{8}} = 8.$
Step 4. $C = \frac{π}{2},$ so the phase shift is $\frac{C}{B} = \frac{\frac{π}{2}}{\frac{π}{8}} = 4.$
Step 5. We draw $f (x) = 4 \tan (\frac{π}{8} x - \frac{π}{2}) - 2.$
Step 6-7. Three points we can use to guide the graph are $(6, 2), (8, - 2),$ and $(10, - 6) .$ We use the reciprocal relationship of tangent and cotangent to draw $f (x) = 4 \cot (\frac{π}{8} x - \frac{π}{2}) - 2.$
Step 8. The vertical asymptotes are $x = 4$ and $x = 12.$

The graph is shown in [link] .

A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12. — One period of a modified cotangent function

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Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

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out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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