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Plotting a function graph/ tan(4*x)

Graphing y = tan(4*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(4*x)
f(x)=tan⁡(4x)f{\left(x \right)} = \tan{\left(4 x \right)} 
f = tan(4*x)
The graph of the function
02468-8-6-4-2-1010-100100
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⁡(4x)=0\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=65.9734457253857x_{1} = 65.9734457253857
x2=2.35619449019234x_{2} = 2.35619449019234
x3=−23.5619449019235x_{3} = -23.5619449019235
x4=−29.845130209103x_{4} = -29.845130209103
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=−11.7809724509617x_{6} = -11.7809724509617
x7=11.7809724509617x_{7} = 11.7809724509617
x8=21.9911485751286x_{8} = 21.9911485751286
x9=46.3384916404494x_{9} = 46.3384916404494
x10=−15.707963267949x_{10} = -15.707963267949
x11=42.4115008234622x_{11} = 42.4115008234622
x12=40.0553063332699x_{12} = 40.0553063332699
x13=−32.2013246992954x_{13} = -32.2013246992954
x14=−25.9181393921158x_{14} = -25.9181393921158
x15=36.1283155162826x_{15} = 36.1283155162826
x16=33.7721210260903x_{16} = 33.7721210260903
x17=55.7632696012188x_{17} = 55.7632696012188
x18=6.28318530717959x_{18} = 6.28318530717959
x19=−80.1106126665397x_{19} = -80.1106126665397
x20=86.3937979737193x_{20} = 86.3937979737193
x21=−18.0641577581413x_{21} = -18.0641577581413
x22=64.4026493985908x_{22} = 64.4026493985908
x23=−69.9004365423729x_{23} = -69.9004365423729
x24=−95.8185759344887x_{24} = -95.8185759344887
x25=28.2743338823081x_{25} = 28.2743338823081
x26=18.0641577581413x_{26} = 18.0641577581413
x27=−85.6083998103219x_{27} = -85.6083998103219
x28=−1.5707963267949x_{28} = -1.5707963267949
x29=47.9092879672443x_{29} = 47.9092879672443
x30=76.1836218495525x_{30} = 76.1836218495525
x31=−62.0464549083984x_{31} = -62.0464549083984
x32=73.8274273593601x_{32} = 73.8274273593601
x33=32.2013246992954x_{33} = 32.2013246992954
x34=62.0464549083984x_{34} = 62.0464549083984
x35=−33.7721210260903x_{35} = -33.7721210260903
x36=−63.6172512351933x_{36} = -63.6172512351933
x37=−10.2101761241668x_{37} = -10.2101761241668
x38=−47.9092879672443x_{38} = -47.9092879672443
x39=10.2101761241668x_{39} = 10.2101761241668
x40=−5.49778714378214x_{40} = -5.49778714378214
x41=72.2566310325652x_{41} = 72.2566310325652
x42=24.3473430653209x_{42} = 24.3473430653209
x43=−51.8362787842316x_{43} = -51.8362787842316
x44=58.1194640914112x_{44} = 58.1194640914112
x45=−76.1836218495525x_{45} = -76.1836218495525
x46=−93.4623814442964x_{46} = -93.4623814442964
x47=−73.8274273593601x_{47} = -73.8274273593601
x48=51.8362787842316x_{48} = 51.8362787842316
x49=78.5398163397448x_{49} = 78.5398163397448
x50=−87.9645943005142x_{50} = -87.9645943005142
x51=−37.6991118430775x_{51} = -37.6991118430775
x52=−43.9822971502571x_{52} = -43.9822971502571
x53=82.4668071567321x_{53} = 82.4668071567321
x54=29.845130209103x_{54} = 29.845130209103
x55=−45.553093477052x_{55} = -45.553093477052
x56=98.174770424681x_{56} = 98.174770424681
x57=−54.1924732744239x_{57} = -54.1924732744239
x58=80.1106126665397x_{58} = 80.1106126665397
x59=−58.1194640914112x_{59} = -58.1194640914112
x60=−98.174770424681x_{60} = -98.174770424681
x61=38.484510006475x_{61} = 38.484510006475
x62=−36.1283155162826x_{62} = -36.1283155162826
x63=−81.6814089933346x_{63} = -81.6814089933346
x64=−49.4800842940392x_{64} = -49.4800842940392
x65=−65.9734457253857x_{65} = -65.9734457253857
x66=0x_{66} = 0
x67=−27.4889357189107x_{67} = -27.4889357189107
x68=−84.037603483527x_{68} = -84.037603483527
x69=−41.6261026600648x_{69} = -41.6261026600648
x70=−67.5442420521806x_{70} = -67.5442420521806
x71=−91.8915851175014x_{71} = -91.8915851175014
x72=43.9822971502571x_{72} = 43.9822971502571
x73=60.4756585816035x_{73} = 60.4756585816035
x74=25.9181393921158x_{74} = 25.9181393921158
x75=100.530964914873x_{75} = 100.530964914873
x76=−7.85398163397448x_{76} = -7.85398163397448
x77=−55.7632696012188x_{77} = -55.7632696012188
x78=−19.6349540849362x_{78} = -19.6349540849362
x79=91.8915851175014x_{79} = 91.8915851175014
x80=7.85398163397448x_{80} = 7.85398163397448
x81=−14.1371669411541x_{81} = -14.1371669411541
x82=50.2654824574367x_{82} = 50.2654824574367
x83=94.2477796076938x_{83} = 94.2477796076938
x84=−3.92699081698724x_{84} = -3.92699081698724
x85=−77.7544181763474x_{85} = -77.7544181763474
x86=−59.6902604182061x_{86} = -59.6902604182061
x87=90.3207887907066x_{87} = 90.3207887907066
x88=−40.0553063332699x_{88} = -40.0553063332699
x89=3.92699081698724x_{89} = 3.92699081698724
x90=14.1371669411541x_{90} = 14.1371669411541
x91=69.9004365423729x_{91} = 69.9004365423729
x92=20.4203522483337x_{92} = 20.4203522483337
x93=54.1924732744239x_{93} = 54.1924732744239
x94=95.8185759344887x_{94} = 95.8185759344887
x95=−89.5353906273091x_{95} = -89.5353906273091
x96=87.9645943005142x_{96} = 87.9645943005142
x97=84.037603483527x_{97} = 84.037603483527
x98=68.329640215578x_{98} = 68.329640215578
x99=−71.4712328691678x_{99} = -71.4712328691678
x100=−99.7455667514759x_{100} = -99.7455667514759
x101=16.4933614313464x_{101} = 16.4933614313464
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(4*x).
tan⁡(0⋅4)\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
4tan⁡2(4x)+4=04 \tan^{2}{\left(4 x \right)} + 4 = 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
32(tan⁡2(4x)+1)tan⁡(4x)=032 \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
lim⁡x→−∞tan⁡(4x)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(4 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⁡(4x)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(4 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(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→−∞(tan⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)
True

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