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  • Graphing y =:
  • x^2/1-x
  • 2x^2-5x
  • x+x
  • /x/

  • Identical expressions

  • tg(x+(2pi)/ three)
  • tg(x plus (2 Pi ) divide by 3)
  • tg(x plus (2 Pi ) divide by three)
  • tgx+2pi/3
  • tg(x+(2pi) divide by 3)

  • Similar expressions

  • tg(x-(2pi)/3)
Plotting a function graph/ tg(x+(2pi)/3)

Graphing y = tg(x+(2pi)/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /    2*pi\
f(x) = tan|x + ----|
          \     3  /
f(x)=tan⁡(x+2π3)f{\left(x \right)} = \tan{\left(x + \frac{2 \pi}{3} \right)} 
f = tan(x + (2*pi)/3)
The graph of the function
02468-8-6-4-2-1010-500500
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:
tan⁡(x+2π3)=0\tan{\left(x + \frac{2 \pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
Numerical solution
x1=16.7551608191456x_{1} = 16.7551608191456
x2=−77.4926187885482x_{2} = -77.4926187885482
x3=−14.6607657167524x_{3} = -14.6607657167524
x4=32.4631240870945x_{4} = 32.4631240870945
x5=1.0471975511966x_{5} = 1.0471975511966
x6=13.6135681655558x_{6} = 13.6135681655558
x7=−86.9173967493176x_{7} = -86.9173967493176
x8=29.3215314335047x_{8} = 29.3215314335047
x9=82.7286065445312x_{9} = 82.7286065445312
x10=−71.2094334813686x_{10} = -71.2094334813686
x11=−2.0943951023932x_{11} = -2.0943951023932
x12=−99.4837673636768x_{12} = -99.4837673636768
x13=−68.0678408277789x_{13} = -68.0678408277789
x14=73.3038285837618x_{14} = 73.3038285837618
x15=−96.342174710087x_{15} = -96.342174710087
x16=41.8879020478639x_{16} = 41.8879020478639
x17=70.162235930172x_{17} = 70.162235930172
x18=51.3126800086333x_{18} = 51.3126800086333
x19=−5.23598775598299x_{19} = -5.23598775598299
x20=−36.6519142918809x_{20} = -36.6519142918809
x21=54.4542726622231x_{21} = 54.4542726622231
x22=−55.5014702134197x_{22} = -55.5014702134197
x23=95.2949771588904x_{23} = 95.2949771588904
x24=10.471975511966x_{24} = 10.471975511966
x25=60.7374579694027x_{25} = 60.7374579694027
x26=−39.7935069454707x_{26} = -39.7935069454707
x27=−46.0766922526503x_{27} = -46.0766922526503
x28=−64.9262481741891x_{28} = -64.9262481741891
x29=−27.2271363311115x_{29} = -27.2271363311115
x30=−90.0589894029074x_{30} = -90.0589894029074
x31=−52.3598775598299x_{31} = -52.3598775598299
x32=48.1710873550435x_{32} = 48.1710873550435
x33=−80.634211442138x_{33} = -80.634211442138
x34=−49.2182849062401x_{34} = -49.2182849062401
x35=38.7463093942741x_{35} = 38.7463093942741
x36=67.0206432765823x_{36} = 67.0206432765823
x37=63.8790506229925x_{37} = 63.8790506229925
x38=19.8967534727354x_{38} = 19.8967534727354
x39=−58.6430628670095x_{39} = -58.6430628670095
x40=57.5958653158129x_{40} = 57.5958653158129
x41=85.870199198121x_{41} = 85.870199198121
x42=−11.5191730631626x_{42} = -11.5191730631626
x43=26.1799387799149x_{43} = 26.1799387799149
x44=35.6047167406843x_{44} = 35.6047167406843
x45=−33.5103216382911x_{45} = -33.5103216382911
x46=−93.2005820564972x_{46} = -93.2005820564972
x47=−24.0855436775217x_{47} = -24.0855436775217
x48=23.0383461263252x_{48} = 23.0383461263252
x49=−61.7846555205993x_{49} = -61.7846555205993
x50=45.0294947014537x_{50} = 45.0294947014537
x51=−83.7758040957278x_{51} = -83.7758040957278
x52=−74.3510261349584x_{52} = -74.3510261349584
x53=−30.3687289847013x_{53} = -30.3687289847013
x54=92.1533845053006x_{54} = 92.1533845053006
x55=4.18879020478639x_{55} = 4.18879020478639
x56=89.0117918517108x_{56} = 89.0117918517108
x57=7.33038285837618x_{57} = 7.33038285837618
x58=76.4454212373516x_{58} = 76.4454212373516
x59=−17.8023583703422x_{59} = -17.8023583703422
x60=101.57816246607x_{60} = 101.57816246607
x61=79.5870138909414x_{61} = 79.5870138909414
x62=−8.37758040957278x_{62} = -8.37758040957278
x63=−20.943951023932x_{63} = -20.943951023932
x64=98.4365698124802x_{64} = 98.4365698124802
x65=−42.9350995990605x_{65} = -42.9350995990605
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x + (2*pi)/3).
tan⁡(2π3)\tan{\left(\frac{2 \pi}{3} \right)}
The result:
f(0)=−3f{\left(0 \right)} = - \sqrt{3}
The point:
(0, -sqrt(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
tan⁡2(x+2π3)+1=0\tan^{2}{\left(x + \frac{2 \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+π6)+1)cot⁡(x+π6)=0- 2 \left(\cot^{2}{\left(x + \frac{\pi}{6} \right)} + 1\right) \cot{\left(x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=π3x_{1} = \frac{\pi}{3}

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
[π3,∞)\left[\frac{\pi}{3}, \infty\right)
Convex at the intervals
(−∞,π3]\left(-\infty, \frac{\pi}{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞tan⁡(x+2π3)y = \lim_{x \to -\infty} \tan{\left(x + \frac{2 \pi}{3} \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞tan⁡(x+2π3)y = \lim_{x \to \infty} \tan{\left(x + \frac{2 \pi}{3} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x + (2*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→−∞(tan⁡(x+2π3)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x + \frac{2 \pi}{3} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x+2π3)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x + \frac{2 \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:
tan⁡(x+2π3)=−tan⁡(x−2π3)\tan{\left(x + \frac{2 \pi}{3} \right)} = - \tan{\left(x - \frac{2 \pi}{3} \right)}
- No
tan⁡(x+2π3)=tan⁡(x−2π3)\tan{\left(x + \frac{2 \pi}{3} \right)} = \tan{\left(x - \frac{2 \pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
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