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sin(2*x)*cos(2*x)
Plotting a function graph/ sin(2*x)*cos(2*x)

Graphing y = sin(2*x)*cos(2*x)

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = sin(2*x)*cos(2*x)
f(x)=sin⁡(2x)cos⁡(2x)f{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(2 x \right)} 
f = sin(2*x)*cos(2*x)
The graph of the function
05-60-55-50-45-40-35-30-25-20-15-10-5101-1
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⁡(2x)cos⁡(2x)=0\sin{\left(2 x \right)} \cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π4x_{2} = \frac{\pi}{4}
x3=π2x_{3} = \frac{\pi}{2}
x4=3π4x_{4} = \frac{3 \pi}{4}
Numerical solution
x1=−63.6172512351933x_{1} = -63.6172512351933
x2=−29.845130209103x_{2} = -29.845130209103
x3=−59.6902604182061x_{3} = -59.6902604182061
x4=86.3937979737193x_{4} = 86.3937979737193
x5=−11.7809724509617x_{5} = -11.7809724509617
x6=90.3207887907066x_{6} = 90.3207887907066
x7=−87.9645943005142x_{7} = -87.9645943005142
x8=−14.1371669411541x_{8} = -14.1371669411541
x9=6.28318530717959x_{9} = 6.28318530717959
x10=18.0641577581413x_{10} = 18.0641577581413
x11=−37.6991118430775x_{11} = -37.6991118430775
x12=−62.0464549083984x_{12} = -62.0464549083984
x13=91.8915851175014x_{13} = 91.8915851175014
x14=83.2522053201295x_{14} = 83.2522053201295
x15=36.1283155162826x_{15} = 36.1283155162826
x16=0x_{16} = 0
x17=−1.5707963267949x_{17} = -1.5707963267949
x18=62.0464549083984x_{18} = 62.0464549083984
x19=40.0553063332699x_{19} = 40.0553063332699
x20=25.9181393921158x_{20} = 25.9181393921158
x21=54.9778714378214x_{21} = 54.9778714378214
x22=87.9645943005142x_{22} = 87.9645943005142
x23=68.329640215578x_{23} = 68.329640215578
x24=−45.553093477052x_{24} = -45.553093477052
x25=2.35619449019234x_{25} = 2.35619449019234
x26=20.4203522483337x_{26} = 20.4203522483337
x27=−69.9004365423729x_{27} = -69.9004365423729
x28=84.037603483527x_{28} = 84.037603483527
x29=−3.92699081698724x_{29} = -3.92699081698724
x30=46.3384916404494x_{30} = 46.3384916404494
x31=−65.9734457253857x_{31} = -65.9734457253857
x32=−99.7455667514759x_{32} = -99.7455667514759
x33=−91.8915851175014x_{33} = -91.8915851175014
x34=65.9734457253857x_{34} = 65.9734457253857
x35=−109.170344712245x_{35} = -109.170344712245
x36=−33.7721210260903x_{36} = -33.7721210260903
x37=28.2743338823081x_{37} = 28.2743338823081
x38=−25.9181393921158x_{38} = -25.9181393921158
x39=−51.8362787842316x_{39} = -51.8362787842316
x40=181.426975744811x_{40} = 181.426975744811
x41=−15.707963267949x_{41} = -15.707963267949
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=−95.8185759344887x_{43} = -95.8185759344887
x44=32.2013246992954x_{44} = 32.2013246992954
x45=3.92699081698724x_{45} = 3.92699081698724
x46=−47.9092879672443x_{46} = -47.9092879672443
x47=−54.1924732744239x_{47} = -54.1924732744239
x48=14.1371669411541x_{48} = 14.1371669411541
x49=−84.037603483527x_{49} = -84.037603483527
x50=95.8185759344887x_{50} = 95.8185759344887
x51=−7.85398163397448x_{51} = -7.85398163397448
x52=−21.9911485751286x_{52} = -21.9911485751286
x53=64.4026493985908x_{53} = 64.4026493985908
x54=−85.6083998103219x_{54} = -85.6083998103219
x55=−10.9955742875643x_{55} = -10.9955742875643
x56=21.9911485751286x_{56} = 21.9911485751286
x57=10.9955742875643x_{57} = 10.9955742875643
x58=58.1194640914112x_{58} = 58.1194640914112
x59=−32.2013246992954x_{59} = -32.2013246992954
x60=43.9822971502571x_{60} = 43.9822971502571
x61=42.4115008234622x_{61} = 42.4115008234622
x62=54.1924732744239x_{62} = 54.1924732744239
x63=29.845130209103x_{63} = 29.845130209103
x64=94.2477796076938x_{64} = 94.2477796076938
x65=17.2787595947439x_{65} = 17.2787595947439
x66=−55.7632696012188x_{66} = -55.7632696012188
x67=69.9004365423729x_{67} = 69.9004365423729
x68=10.2101761241668x_{68} = 10.2101761241668
x69=−73.8274273593601x_{69} = -73.8274273593601
x70=−81.6814089933346x_{70} = -81.6814089933346
x71=80.1106126665397x_{71} = 80.1106126665397
x72=24.3473430653209x_{72} = 24.3473430653209
x73=−77.7544181763474x_{73} = -77.7544181763474
x74=−69.1150383789755x_{74} = -69.1150383789755
x75=−18.0641577581413x_{75} = -18.0641577581413
x76=−23.5619449019235x_{76} = -23.5619449019235
x77=98.174770424681x_{77} = 98.174770424681
x78=−98.174770424681x_{78} = -98.174770424681
x79=−64.4026493985908x_{79} = -64.4026493985908
x80=−80.1106126665397x_{80} = -80.1106126665397
x81=−19.6349540849362x_{81} = -19.6349540849362
x82=76.1836218495525x_{82} = 76.1836218495525
x83=7.85398163397448x_{83} = 7.85398163397448
x84=−76.1836218495525x_{84} = -76.1836218495525
x85=−40.0553063332699x_{85} = -40.0553063332699
x86=51.8362787842316x_{86} = 51.8362787842316
x87=−58.1194640914112x_{87} = -58.1194640914112
x88=47.9092879672443x_{88} = 47.9092879672443
x89=−41.6261026600648x_{89} = -41.6261026600648
x90=−43.9822971502571x_{90} = -43.9822971502571
x91=72.2566310325652x_{91} = 72.2566310325652
x92=73.8274273593601x_{92} = 73.8274273593601
x93=50.2654824574367x_{93} = 50.2654824574367
x94=−88.7499924639117x_{94} = -88.7499924639117
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)*cos(2*x).
sin⁡(2⋅0)cos⁡(2⋅0)\sin{\left(2 \cdot 0 \right)} \cos{\left(2 \cdot 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
−2sin⁡2(2x)+2cos⁡2(2x)=0- 2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−π8x_{1} = - \frac{\pi}{8}
x2=π8x_{2} = \frac{\pi}{8}
The values of the extrema at the points:
 -pi        
(----, -1/2)
  8         

 pi      
(--, 1/2)
 8


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=−π8x_{1} = - \frac{\pi}{8}
Maxima of the function at points:
x1=π8x_{1} = \frac{\pi}{8}
Decreasing at intervals
[−π8,π8]\left[- \frac{\pi}{8}, \frac{\pi}{8}\right]
Increasing at intervals
(−∞,−π8]∪[π8,∞)\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{\pi}{8}, \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
−16sin⁡(2x)cos⁡(2x)=0- 16 \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=π2x_{3} = \frac{\pi}{2}
x4=3π4x_{4} = \frac{3 \pi}{4}

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
[3π4,∞)\left[\frac{3 \pi}{4}, \infty\right)
Convex at the intervals
(−∞,π4]∪[π2,3π4]\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{\pi}{2}, \frac{3 \pi}{4}\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⁡(2x)cos⁡(2x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(2x)cos⁡(2x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x)*cos(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos{\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⁡(2x)cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos{\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⁡(2x)cos⁡(2x)=−sin⁡(2x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(2 x \right)} = - \sin{\left(2 x \right)} \cos{\left(2 x \right)}
- No
sin⁡(2x)cos⁡(2x)=sin⁡(2x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(2 x \right)} = \sin{\left(2 x \right)} \cos{\left(2 x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(2*x)*cos(2*x)
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