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Plotting a function graph/ 2*tan(4*x)

Graphing y = 2*tan(4*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 2*tan(4*x)
f(x)=2tan⁡(4x)f{\left(x \right)} = 2 \tan{\left(4 x \right)} 
f = 2*tan(4*x)
The graph of the function
02468-8-6-4-2-1010-250250
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:
2tan⁡(4x)=02 \tan{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=78.5398163397448x_{1} = 78.5398163397448
x2=−5.49778714378214x_{2} = -5.49778714378214
x3=−65.9734457253857x_{3} = -65.9734457253857
x4=−40.0553063332699x_{4} = -40.0553063332699
x5=−15.707963267949x_{5} = -15.707963267949
x6=86.3937979737193x_{6} = 86.3937979737193
x7=50.2654824574367x_{7} = 50.2654824574367
x8=84.037603483527x_{8} = 84.037603483527
x9=42.4115008234622x_{9} = 42.4115008234622
x10=54.1924732744239x_{10} = 54.1924732744239
x11=73.8274273593601x_{11} = 73.8274273593601
x12=11.7809724509617x_{12} = 11.7809724509617
x13=68.329640215578x_{13} = 68.329640215578
x14=−1.5707963267949x_{14} = -1.5707963267949
x15=−58.1194640914112x_{15} = -58.1194640914112
x16=−54.1924732744239x_{16} = -54.1924732744239
x17=−51.8362787842316x_{17} = -51.8362787842316
x18=−47.9092879672443x_{18} = -47.9092879672443
x19=−18.0641577581413x_{19} = -18.0641577581413
x20=90.3207887907066x_{20} = 90.3207887907066
x21=7.85398163397448x_{21} = 7.85398163397448
x22=58.1194640914112x_{22} = 58.1194640914112
x23=−67.5442420521806x_{23} = -67.5442420521806
x24=−59.6902604182061x_{24} = -59.6902604182061
x25=21.9911485751286x_{25} = 21.9911485751286
x26=−84.037603483527x_{26} = -84.037603483527
x27=6.28318530717959x_{27} = 6.28318530717959
x28=−69.9004365423729x_{28} = -69.9004365423729
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=−19.6349540849362x_{30} = -19.6349540849362
x31=−27.4889357189107x_{31} = -27.4889357189107
x32=−3.92699081698724x_{32} = -3.92699081698724
x33=−55.7632696012188x_{33} = -55.7632696012188
x34=−95.8185759344887x_{34} = -95.8185759344887
x35=−76.1836218495525x_{35} = -76.1836218495525
x36=62.0464549083984x_{36} = 62.0464549083984
x37=−32.2013246992954x_{37} = -32.2013246992954
x38=29.845130209103x_{38} = 29.845130209103
x39=28.2743338823081x_{39} = 28.2743338823081
x40=80.1106126665397x_{40} = 80.1106126665397
x41=94.2477796076938x_{41} = 94.2477796076938
x42=25.9181393921158x_{42} = 25.9181393921158
x43=−77.7544181763474x_{43} = -77.7544181763474
x44=47.9092879672443x_{44} = 47.9092879672443
x45=91.8915851175014x_{45} = 91.8915851175014
x46=−80.1106126665397x_{46} = -80.1106126665397
x47=51.8362787842316x_{47} = 51.8362787842316
x48=−29.845130209103x_{48} = -29.845130209103
x49=24.3473430653209x_{49} = 24.3473430653209
x50=38.484510006475x_{50} = 38.484510006475
x51=40.0553063332699x_{51} = 40.0553063332699
x52=20.4203522483337x_{52} = 20.4203522483337
x53=14.1371669411541x_{53} = 14.1371669411541
x54=98.174770424681x_{54} = 98.174770424681
x55=10.2101761241668x_{55} = 10.2101761241668
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=−23.5619449019235x_{57} = -23.5619449019235
x58=−36.1283155162826x_{58} = -36.1283155162826
x59=55.7632696012188x_{59} = 55.7632696012188
x60=−49.4800842940392x_{60} = -49.4800842940392
x61=36.1283155162826x_{61} = 36.1283155162826
x62=60.4756585816035x_{62} = 60.4756585816035
x63=−81.6814089933346x_{63} = -81.6814089933346
x64=43.9822971502571x_{64} = 43.9822971502571
x65=95.8185759344887x_{65} = 95.8185759344887
x66=−14.1371669411541x_{66} = -14.1371669411541
x67=−99.7455667514759x_{67} = -99.7455667514759
x68=−62.0464549083984x_{68} = -62.0464549083984
x69=−71.4712328691678x_{69} = -71.4712328691678
x70=76.1836218495525x_{70} = 76.1836218495525
x71=3.92699081698724x_{71} = 3.92699081698724
x72=−33.7721210260903x_{72} = -33.7721210260903
x73=18.0641577581413x_{73} = 18.0641577581413
x74=−41.6261026600648x_{74} = -41.6261026600648
x75=0x_{75} = 0
x76=−21.9911485751286x_{76} = -21.9911485751286
x77=−85.6083998103219x_{77} = -85.6083998103219
x78=−89.5353906273091x_{78} = -89.5353906273091
x79=100.530964914873x_{79} = 100.530964914873
x80=69.9004365423729x_{80} = 69.9004365423729
x81=65.9734457253857x_{81} = 65.9734457253857
x82=−45.553093477052x_{82} = -45.553093477052
x83=−63.6172512351933x_{83} = -63.6172512351933
x84=−11.7809724509617x_{84} = -11.7809724509617
x85=−10.2101761241668x_{85} = -10.2101761241668
x86=−7.85398163397448x_{86} = -7.85398163397448
x87=82.4668071567321x_{87} = 82.4668071567321
x88=46.3384916404494x_{88} = 46.3384916404494
x89=64.4026493985908x_{89} = 64.4026493985908
x90=−93.4623814442964x_{90} = -93.4623814442964
x91=−91.8915851175014x_{91} = -91.8915851175014
x92=−73.8274273593601x_{92} = -73.8274273593601
x93=32.2013246992954x_{93} = 32.2013246992954
x94=−98.174770424681x_{94} = -98.174770424681
x95=−43.9822971502571x_{95} = -43.9822971502571
x96=−25.9181393921158x_{96} = -25.9181393921158
x97=16.4933614313464x_{97} = 16.4933614313464
x98=72.2566310325652x_{98} = 72.2566310325652
x99=2.35619449019234x_{99} = 2.35619449019234
x100=33.7721210260903x_{100} = 33.7721210260903
x101=87.9645943005142x_{101} = 87.9645943005142
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*tan(4*x).
2tan⁡(0⋅4)2 \tan{\left(0 \cdot 4 \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
8tan⁡2(4x)+8=08 \tan^{2}{\left(4 x \right)} + 8 = 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
64(tan⁡2(4x)+1)tan⁡(4x)=064 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 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
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(2tan⁡(4x))y = \lim_{x \to -\infty}\left(2 \tan{\left(4 x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(2tan⁡(4x))y = \lim_{x \to \infty}\left(2 \tan{\left(4 x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*tan(4*x), 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→−∞(2tan⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{2 \tan{\left(4 x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(2tan⁡(4x)x)y = x \lim_{x \to \infty}\left(\frac{2 \tan{\left(4 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:
2tan⁡(4x)=−2tan⁡(4x)2 \tan{\left(4 x \right)} = - 2 \tan{\left(4 x \right)}
- No
2tan⁡(4x)=2tan⁡(4x)2 \tan{\left(4 x \right)} = 2 \tan{\left(4 x \right)}
- Yes
so, the function
is
odd
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