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tan(x)
Plotting a function graph/ tan(x)

Graphing y = tan(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(x)
f(x)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} 
f = tan(x)
The graph of the function
0-60-50-40-30-20-1010203040010
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)=0\tan{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−40.8407044966673x_{1} = -40.8407044966673
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=47.1238898038469x_{3} = 47.1238898038469
x4=75.398223686155x_{4} = 75.398223686155
x5=12.5663706143592x_{5} = 12.5663706143592
x6=−53.4070751110265x_{6} = -53.4070751110265
x7=−75.398223686155x_{7} = -75.398223686155
x8=−47.1238898038469x_{8} = -47.1238898038469
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=−59.6902604182061x_{10} = -59.6902604182061
x11=−94.2477796076938x_{11} = -94.2477796076938
x12=−18.8495559215388x_{12} = -18.8495559215388
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=−81.6814089933346x_{14} = -81.6814089933346
x15=6.28318530717959x_{15} = 6.28318530717959
x16=−15.707963267949x_{16} = -15.707963267949
x17=56.5486677646163x_{17} = 56.5486677646163
x18=50.2654824574367x_{18} = 50.2654824574367
x19=78.5398163397448x_{19} = 78.5398163397448
x20=−12.5663706143592x_{20} = -12.5663706143592
x21=72.2566310325652x_{21} = 72.2566310325652
x22=53.4070751110265x_{22} = 53.4070751110265
x23=15.707963267949x_{23} = 15.707963267949
x24=62.8318530717959x_{24} = 62.8318530717959
x25=18.8495559215388x_{25} = 18.8495559215388
x26=−37.6991118430775x_{26} = -37.6991118430775
x27=−31.4159265358979x_{27} = -31.4159265358979
x28=37.6991118430775x_{28} = 37.6991118430775
x29=59.6902604182061x_{29} = 59.6902604182061
x30=0x_{30} = 0
x31=−3.14159265358979x_{31} = -3.14159265358979
x32=69.1150383789755x_{32} = 69.1150383789755
x33=−43.9822971502571x_{33} = -43.9822971502571
x34=−91.106186954104x_{34} = -91.106186954104
x35=−28.2743338823081x_{35} = -28.2743338823081
x36=−72.2566310325652x_{36} = -72.2566310325652
x37=−34.5575191894877x_{37} = -34.5575191894877
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=28.2743338823081x_{39} = 28.2743338823081
x40=40.8407044966673x_{40} = 40.8407044966673
x41=−9.42477796076938x_{41} = -9.42477796076938
x42=94.2477796076938x_{42} = 94.2477796076938
x43=−6.28318530717959x_{43} = -6.28318530717959
x44=9.42477796076938x_{44} = 9.42477796076938
x45=21.9911485751286x_{45} = 21.9911485751286
x46=97.3893722612836x_{46} = 97.3893722612836
x47=81.6814089933346x_{47} = 81.6814089933346
x48=−97.3893722612836x_{48} = -97.3893722612836
x49=43.9822971502571x_{49} = 43.9822971502571
x50=100.530964914873x_{50} = 100.530964914873
x51=25.1327412287183x_{51} = 25.1327412287183
x52=−84.8230016469244x_{52} = -84.8230016469244
x53=−69.1150383789755x_{53} = -69.1150383789755
x54=91.106186954104x_{54} = 91.106186954104
x55=3.14159265358979x_{55} = 3.14159265358979
x56=−78.5398163397448x_{56} = -78.5398163397448
x57=84.8230016469244x_{57} = 84.8230016469244
x58=34.5575191894877x_{58} = 34.5575191894877
x59=−25.1327412287183x_{59} = -25.1327412287183
x60=−87.9645943005142x_{60} = -87.9645943005142
x61=65.9734457253857x_{61} = 65.9734457253857
x62=−100.530964914873x_{62} = -100.530964914873
x63=−50.2654824574367x_{63} = -50.2654824574367
x64=87.9645943005142x_{64} = 87.9645943005142
x65=31.4159265358979x_{65} = 31.4159265358979
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x).
tan⁡(0)\tan{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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)+1=0\tan^{2}{\left(x \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(tan⁡2(x)+1)tan⁡(x)=02 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞tan⁡(x)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞tan⁡(x)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(x)x)=lim⁡x→−∞(tan⁡(x)x)\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x}\right)
lim⁡x→∞(tan⁡(x)x)=lim⁡x→∞(tan⁡(x)x)\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \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)=−tan⁡(x)\tan{\left(x \right)} = - \tan{\left(x \right)}
- No
tan⁡(x)=tan⁡(x)\tan{\left(x \right)} = \tan{\left(x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = tan(x)
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