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  • How to use it?

  • Graphing y =:
  • (x^2-3x+2)/(x+1)
  • x/(1-x^2)
  • y=|x-2|-|x+1|+x-2 y=|x-2|-|x+1|+x-2
  • -1/3x^3+4x

  • Identical expressions

  • tan(four *x)^ two
  • tangent of (4 multiply by x) squared
  • tangent of (four multiply by x) to the power of two
  • tan(4*x)2
  • tan4*x2
  • tan(4*x)²
  • tan(4*x) to the power of 2
  • tan(4x)^2
  • tan(4x)2
  • tan4x2
  • tan4x^2
Plotting a function graph/ tan(4*x)^2

Graphing y = tan(4*x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2     
f(x) = tan (4*x)
f(x)=tan⁡2(4x)f{\left(x \right)} = \tan^{2}{\left(4 x \right)} 
f = tan(4*x)^2
The graph of the function
02468-8-6-4-2-101005000
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⁡2(4x)=0\tan^{2}{\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=−62.0464548538807x_{1} = -62.0464548538807
x2=−36.1283153226971x_{2} = -36.1283153226971
x3=90.320788735349x_{3} = 90.320788735349
x4=6.28318528262228x_{4} = 6.28318528262228
x5=−23.5619451232182x_{5} = -23.5619451232182
x6=87.9645943416767x_{6} = 87.9645943416767
x7=−43.9822971730515x_{7} = -43.9822971730515
x8=82.4668069120011x_{8} = 82.4668069120011
x9=−65.9734457615098x_{9} = -65.9734457615098
x10=50.2654824460375x_{10} = 50.2654824460375
x11=−14.1371667262421x_{11} = -14.1371667262421
x12=−3.92699109852159x_{12} = -3.92699109852159
x13=29.845130469522x_{13} = 29.845130469522
x14=24.347342983671x_{14} = 24.347342983671
x15=−63.6172513615354x_{15} = -63.6172513615354
x16=−29.845130062694x_{16} = -29.845130062694
x17=28.2743338643801x_{17} = 28.2743338643801
x18=−7.85398142254902x_{18} = -7.85398142254902
x19=20.4203520591665x_{19} = 20.4203520591665
x20=94.2477796093471x_{20} = 94.2477796093471
x21=60.4756583546577x_{21} = 60.4756583546577
x22=33.7721212629029x_{22} = 33.7721212629029
x23=72.2566310276665x_{23} = 72.2566310276665
x24=−84.037603439755x_{24} = -84.037603439755
x25=54.1924733267508x_{25} = 54.1924733267508
x26=−33.7721210090001x_{26} = -33.7721210090001
x27=36.128315793064x_{27} = 36.128315793064
x28=−94.2477795727592x_{28} = -94.2477795727592
x29=65.9734457562088x_{29} = 65.9734457562088
x30=−40.0553062675254x_{30} = -40.0553062675254
x31=−5.49778706564407x_{31} = -5.49778706564407
x32=47.9092880853601x_{32} = 47.9092880853601
x33=−45.5530937122474x_{33} = -45.5530937122474
x34=−67.5442422970135x_{34} = -67.5442422970135
x35=21.9911485855647x_{35} = 21.9911485855647
x36=0x_{36} = 0
x37=−81.6814090467211x_{37} = -81.6814090467211
x38=−19.6349541873064x_{38} = -19.6349541873064
x39=−87.9645943524031x_{39} = -87.9645943524031
x40=−77.7544181730488x_{40} = -77.7544181730488
x41=69.900436676108x_{41} = 69.900436676108
x42=18.0641575980835x_{42} = 18.0641575980835
x43=−11.7809724263829x_{43} = -11.7809724263829
x44=38.4845098235765x_{44} = 38.4845098235765
x45=−41.62610277398x_{45} = -41.62610277398
x46=−21.991148586076x_{46} = -21.991148586076
x47=14.1371670560163x_{47} = 14.1371670560163
x48=−59.69026046391x_{48} = -59.69026046391
x49=−58.1194639178135x_{49} = -58.1194639178135
x50=77.7544182462014x_{50} = 77.7544182462014
x51=−55.7632695911703x_{51} = -55.7632695911703
x52=99.7455665024765x_{52} = 99.7455665024765
x53=16.4933613318976x_{53} = 16.4933613318976
x54=91.891585268117x_{54} = 91.891585268117
x55=−37.6991118814587x_{55} = -37.6991118814587
x56=64.4026492408947x_{56} = 64.4026492408947
x57=43.9822971708964x_{57} = 43.9822971708964
x58=7.85398187132473x_{58} = 7.85398187132473
x59=10.2101761568911x_{59} = 10.2101761568911
x60=84.0376034271735x_{60} = 84.0376034271735
x61=98.1747705056141x_{61} = 98.1747705056141
x62=−99.745566754756x_{62} = -99.745566754756
x63=−80.1106125115255x_{63} = -80.1106125115255
x64=62.0464548306542x_{64} = 62.0464548306542
x65=46.3384915681264x_{65} = 46.3384915681264
x66=42.4115006503654x_{66} = 42.4115006503654
x67=51.8362790674845x_{67} = 51.8362790674845
x68=−51.8362786747725x_{68} = -51.8362786747725
x69=−73.8274272737004x_{69} = -73.8274272737004
x70=−1.57079653228711x_{70} = -1.57079653228711
x71=−15.7079632993025x_{71} = -15.7079632993025
x72=−95.8185758659514x_{72} = -95.8185758659514
x73=76.1836219145007x_{73} = 76.1836219145007
x74=32.2013247411198x_{74} = 32.2013247411198
x75=−89.5353908736038x_{75} = -89.5353908736038
x76=−18.0641576805559x_{76} = -18.0641576805559
x77=68.3296401519947x_{77} = 68.3296401519947
x78=55.7632697711785x_{78} = 55.7632697711785
x79=40.0553062249401x_{79} = 40.0553062249401
x80=86.3937978305559x_{80} = 86.3937978305559
x81=2.35619439855232x_{81} = 2.35619439855232
x82=3.92699090715964x_{82} = 3.92699090715964
x83=−85.6083999500216x_{83} = -85.6083999500216
x84=25.918139495754x_{84} = 25.918139495754
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)^2.
tan⁡2(0⋅4)\tan^{2}{\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)tan⁡(4x)=0\left(8 \tan^{2}{\left(4 x \right)} + 8\right) \tan{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(0, 0)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=0x_{1} = 0
The function has no maxima
Decreasing at intervals
[0,∞)\left[0, \infty\right)
Increasing at intervals
(−∞,0]\left(-\infty, 0\right]
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)(3tan⁡2(4x)+1)=032 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(3 \tan^{2}{\left(4 x \right)} + 1\right) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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⁡2(4x)y = \lim_{x \to -\infty} \tan^{2}{\left(4 x \right)}
True

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

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