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

Precalculus

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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 $(−∞, - | A |] \cup [| A |, \infty) .$
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.

Got questions? Get instant answers now!

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

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

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bill

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-24m+3+3mÁ^2

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Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

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Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

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Nandala

3x-12y=18

Kelvin

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A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

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The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

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12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

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When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

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Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

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Sheirtina

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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