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Plotting a function graph/ cot(x+(pi/3))

Graphing y = cot(x+(pi/3))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /    pi\
f(x) = cot|x + --|
          \    3 /
f(x)=cot⁡(x+π3)f{\left(x \right)} = \cot{\left(x + \frac{\pi}{3} \right)} 
f = cot(x + pi/3)
The graph of the function
02468-8-6-4-2-1010-200200
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
cot⁡(x+π3)=0\cot{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
Numerical solution
x1=25.6563400043166x_{1} = 25.6563400043166
x2=66.497044500984x_{2} = 66.497044500984
x3=−62.3082542961976x_{3} = -62.3082542961976
x4=−74.8746249105567x_{4} = -74.8746249105567
x5=−81.1578102177363x_{5} = -81.1578102177363
x6=−49.7418836818384x_{6} = -49.7418836818384
x7=−78.0162175641465x_{7} = -78.0162175641465
x8=82.2050077689329x_{8} = 82.2050077689329
x9=−5.75958653158129x_{9} = -5.75958653158129
x10=97.9129710368819x_{10} = 97.9129710368819
x11=41.3643032722656x_{11} = 41.3643032722656
x12=−100.007366139275x_{12} = -100.007366139275
x13=79.0634151153431x_{13} = 79.0634151153431
x14=72.7802298081635x_{14} = 72.7802298081635
x15=−2.61799387799149x_{15} = -2.61799387799149
x16=22.5147473507269x_{16} = 22.5147473507269
x17=−37.1755130674792x_{17} = -37.1755130674792
x18=60.2138591938044x_{18} = 60.2138591938044
x19=88.4881930761125x_{19} = 88.4881930761125
x20=31.9395253114962x_{20} = 31.9395253114962
x21=−34.0339204138894x_{21} = -34.0339204138894
x22=−65.4498469497874x_{22} = -65.4498469497874
x23=19.3731546971371x_{23} = 19.3731546971371
x24=−87.4409955249159x_{24} = -87.4409955249159
x25=−46.6002910282486x_{25} = -46.6002910282486
x26=50.789081233035x_{26} = 50.789081233035
x27=−24.60914245312x_{27} = -24.60914245312
x28=−30.8923277602996x_{28} = -30.8923277602996
x29=53.9306738866248x_{29} = 53.9306738866248
x30=−84.2994028713261x_{30} = -84.2994028713261
x31=−71.733032256967x_{31} = -71.733032256967
x32=101.054563690472x_{32} = 101.054563690472
x33=35.081117965086x_{33} = 35.081117965086
x34=3.66519142918809x_{34} = 3.66519142918809
x35=63.3554518473942x_{35} = 63.3554518473942
x36=−21.4675497995303x_{36} = -21.4675497995303
x37=9.94837673636768x_{37} = 9.94837673636768
x38=16.2315620435473x_{38} = 16.2315620435473
x39=−59.1666616426078x_{39} = -59.1666616426078
x40=91.6297857297023x_{40} = 91.6297857297023
x41=57.0722665402146x_{41} = 57.0722665402146
x42=44.5058959258554x_{42} = 44.5058959258554
x43=13.0899693899575x_{43} = 13.0899693899575
x44=−68.5914396033772x_{44} = -68.5914396033772
x45=−15.1843644923507x_{45} = -15.1843644923507
x46=28.7979326579064x_{46} = 28.7979326579064
x47=−8.90117918517108x_{47} = -8.90117918517108
x48=0.523598775598299x_{48} = 0.523598775598299
x49=−12.0427718387609x_{49} = -12.0427718387609
x50=85.3466004225227x_{50} = 85.3466004225227
x51=−40.317105721069x_{51} = -40.317105721069
x52=38.2227106186758x_{52} = 38.2227106186758
x53=6.80678408277789x_{53} = 6.80678408277789
x54=47.6474885794452x_{54} = 47.6474885794452
x55=−43.4586983746588x_{55} = -43.4586983746588
x56=−93.7241808320955x_{56} = -93.7241808320955
x57=69.6386371545737x_{57} = 69.6386371545737
x58=−18.3259571459405x_{58} = -18.3259571459405
x59=−56.025068989018x_{59} = -56.025068989018
x60=75.9218224617533x_{60} = 75.9218224617533
x61=−27.7507351067098x_{61} = -27.7507351067098
x62=−52.8834763354282x_{62} = -52.8834763354282
x63=−96.8657734856853x_{63} = -96.8657734856853
x64=−90.5825881785057x_{64} = -90.5825881785057
x65=94.7713783832921x_{65} = 94.7713783832921
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(x + pi/3).
cot⁡(π3)\cot{\left(\frac{\pi}{3} \right)}
The result:
f(0)=33f{\left(0 \right)} = \frac{\sqrt{3}}{3}
The point:
(0, sqrt(3)/3)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−cot⁡2(x+π3)−1=0- \cot^{2}{\left(x + \frac{\pi}{3} \right)} - 1 = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(cot⁡2(x+π3)+1)cot⁡(x+π3)=02 \left(\cot^{2}{\left(x + \frac{\pi}{3} \right)} + 1\right) \cot{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,π6]\left(-\infty, \frac{\pi}{6}\right]
Convex at the intervals
[π6,∞)\left[\frac{\pi}{6}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cot⁡(x+π3)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(x + \frac{\pi}{3} \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−cot⁡(∞)y = - \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(x+π3)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(x + \frac{\pi}{3} \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x + pi/3), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(x+π3)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x + \frac{\pi}{3} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x+π3)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x + \frac{\pi}{3} \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
cot⁡(x+π3)=−cot⁡(x−π3)\cot{\left(x + \frac{\pi}{3} \right)} = - \cot{\left(x - \frac{\pi}{3} \right)}
- No
cot⁡(x+π3)=cot⁡(x−π3)\cot{\left(x + \frac{\pi}{3} \right)} = \cot{\left(x - \frac{\pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
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