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Plotting a function graph/ sin^2(2x)

Graphing y = sin^2(2x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2     
f(x) = sin (2*x)
f(x)=sin⁡2(2x)f{\left(x \right)} = \sin^{2}{\left(2 x \right)} 
f = sin(2*x)^2
The graph of the function
0-800-600-400-200200400600800-1000100002
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:
sin⁡2(2x)=0\sin^{2}{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=−29.8451301000724x_{1} = -29.8451301000724
x2=89.5353906153414x_{2} = 89.5353906153414
x3=78.5398162225044x_{3} = 78.5398162225044
x4=37.6991119665793x_{4} = 37.6991119665793
x5=48.6946860958663x_{5} = 48.6946860958663
x6=−14.1371668484631x_{6} = -14.1371668484631
x7=−58.1194645366003x_{7} = -58.1194645366003
x8=−17.2787595621355x_{8} = -17.2787595621355
x9=−94.247779486083x_{9} = -94.247779486083
x10=−21.9911485864927x_{10} = -21.9911485864927
x11=29.8451303084991x_{11} = 29.8451303084991
x12=59.690260541069x_{12} = 59.690260541069
x13=−1.57079642013166x_{13} = -1.57079642013166
x14=−97.3893723711949x_{14} = -97.3893723711949
x15=21.9911485851564x_{15} = 21.9911485851564
x16=−80.1106125854791x_{16} = -80.1106125854791
x17=28.2743338652921x_{17} = 28.2743338652921
x18=51.8362788866811x_{18} = 51.8362788866811
x19=−7.85398150696156x_{19} = -7.85398150696156
x20=4.71238898608896x_{20} = 4.71238898608896
x21=72.2566310277248x_{21} = 72.2566310277248
x22=7.85398173011892x_{22} = 7.85398173011892
x23=−73.8274272808521x_{23} = -73.8274272808521
x24=−72.2566309100272x_{24} = -72.2566309100272
x25=−51.8362786915081x_{25} = -51.8362786915081
x26=26.7035375390573x_{26} = 26.7035375390573
x27=56.5486676469942x_{27} = 56.5486676469942
x28=−42.4115007432387x_{28} = -42.4115007432387
x29=70.6858346557926x_{29} = 70.6858346557926
x30=−75.3982237985682x_{30} = -75.3982237985682
x31=6.28318528443138x_{31} = 6.28318528443138
x32=86.3937978937855x_{32} = 86.3937978937855
x33=15.7079633917898x_{33} = 15.7079633917898
x34=34.5575190717885x_{34} = 34.5575190717885
x35=−81.6814090370675x_{35} = -81.6814090370675
x36=94.247779609353x_{36} = 94.247779609353
x37=14.1371670778185x_{37} = 14.1371670778185
x38=117.809724442492x_{38} = 117.809724442492
x39=−1.57079626356835x_{39} = -1.57079626356835
x40=36.1283154718409x_{40} = 36.1283154718409
x41=1.57079638652515x_{41} = 1.57079638652515
x42=−31.4159266517141x_{42} = -31.4159266517141
x43=100.530964798296x_{43} = 100.530964798296
x44=−28.2743337586152x_{44} = -28.2743337586152
x45=−15.7079632962205x_{45} = -15.7079632962205
x46=72.256630710694x_{46} = 72.256630710694
x47=−36.128315427252x_{47} = -36.128315427252
x48=−95.818575868455x_{48} = -95.818575868455
x49=−97.3893725907902x_{49} = -97.3893725907902
x50=−6.28318518328035x_{50} = -6.28318518328035
x51=−37.6991118766796x_{51} = -37.6991118766796
x52=−83.2522051669813x_{52} = -83.2522051669813
x53=−50.2654823342013x_{53} = -50.2654823342013
x54=42.4115007365289x_{54} = 42.4115007365289
x55=−53.4070752253874x_{55} = -53.4070752253874
x56=−43.9822971747455x_{56} = -43.9822971747455
x57=−20.4203521774723x_{57} = -20.4203521774723
x58=81.6814091152362x_{58} = 81.6814091152362
x59=45.5530935075531x_{59} = 45.5530935075531
x60=−23.5619449982306x_{60} = -23.5619449982306
x61=12.5663704969137x_{61} = 12.5663704969137
x62=67.5442420634706x_{62} = 67.5442420634706
x63=50.2654824463816x_{63} = 50.2654824463816
x64=−58.1194640062544x_{64} = -58.1194640062544
x65=−86.3937978789102x_{65} = -86.3937978789102
x66=−39.2699081045218x_{66} = -39.2699081045218
x67=20.4203521581227x_{67} = 20.4203521581227
x68=−67.5442421539445x_{68} = -67.5442421539445
x69=−59.6902604569585x_{69} = -59.6902604569585
x70=0x_{70} = 0
x71=−65.9734457653935x_{71} = -65.9734457653935
x72=−61.2610566398387x_{72} = -61.2610566398387
x73=73.8274274646672x_{73} = 73.8274274646672
x74=−100.530965206253x_{74} = -100.530965206253
x75=95.8185760424586x_{75} = 95.8185760424586
x76=−89.5353907315491x_{76} = -89.5353907315491
x77=64.4026493150839x_{77} = 64.4026493150839
x78=80.1106131511482x_{78} = 80.1106131511482
x79=65.9734457525462x_{79} = 65.9734457525462
x80=92.6769832182628x_{80} = 92.6769832182628
x81=87.9645943351391x_{81} = 87.9645943351391
x82=−64.402649310466x_{82} = -64.402649310466
x83=−45.5530935761698x_{83} = -45.5530935761698
x84=−9.42477807759933x_{84} = -9.42477807759933
x85=43.9822971692691x_{85} = 43.9822971692691
x86=95.8185756842062x_{86} = 95.8185756842062
x87=23.5619449483644x_{87} = 23.5619449483644
x88=−87.9645943594276x_{88} = -87.9645943594276
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2*x)^2.
sin⁡2(0⋅2)\sin^{2}{\left(0 \cdot 2 \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
4sin⁡(2x)cos⁡(2x)=04 \sin{\left(2 x \right)} \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π4x_{2} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
The values of the extrema at the points:
(0, 0)

 -pi     
(----, 1)
  4      

 pi    
(--, 1)
 4


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
Maxima of the function at points:
x1=−π4x_{1} = - \frac{\pi}{4}
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,−π4]∪[0,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[π4,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{4}, \infty\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
8(−sin⁡2(2x)+cos⁡2(2x))=08 \left(- \sin^{2}{\left(2 x \right)} + \cos^{2}{\left(2 x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π8x_{1} = - \frac{\pi}{8}
x2=π8x_{2} = \frac{\pi}{8}

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
[−π8,π8]\left[- \frac{\pi}{8}, \frac{\pi}{8}\right]
Convex at the intervals
(−∞,−π8]∪[π8,∞)\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{\pi}{8}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞sin⁡2(2x)=⟨0,1⟩\lim_{x \to -\infty} \sin^{2}{\left(2 x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞sin⁡2(2x)=⟨0,1⟩\lim_{x \to \infty} \sin^{2}{\left(2 x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡2(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
sin⁡2(2x)=sin⁡2(2x)\sin^{2}{\left(2 x \right)} = \sin^{2}{\left(2 x \right)}
- Yes
sin⁡2(2x)=−sin⁡2(2x)\sin^{2}{\left(2 x \right)} = - \sin^{2}{\left(2 x \right)}
- No
so, the function
is
even
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