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

Graphing y = 2*sin(2*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(2*x)
f(x)=2sin⁡(2x)f{\left(x \right)} = 2 \sin{\left(2 x \right)} 
f = 2*sin(2*x)
The graph of the function
0-101020304050607080901001105-5
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:
2sin⁡(2x)=02 \sin{\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=113.097335529233x_{1} = 113.097335529233
x2=−15.707963267949x_{2} = -15.707963267949
x3=81.6814089933346x_{3} = 81.6814089933346
x4=−86.3937979737193x_{4} = -86.3937979737193
x5=−6.28318530717959x_{5} = -6.28318530717959
x6=−23.5619449019235x_{6} = -23.5619449019235
x7=70.6858347057703x_{7} = 70.6858347057703
x8=−51.8362787842316x_{8} = -51.8362787842316
x9=−45.553093477052x_{9} = -45.553093477052
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=−72.2566310325652x_{11} = -72.2566310325652
x12=−75.398223686155x_{12} = -75.398223686155
x13=80.1106126665397x_{13} = 80.1106126665397
x14=−119.380520836412x_{14} = -119.380520836412
x15=−50.2654824574367x_{15} = -50.2654824574367
x16=−43.9822971502571x_{16} = -43.9822971502571
x17=87.9645943005142x_{17} = 87.9645943005142
x18=−40.8407044966673x_{18} = -40.8407044966673
x19=72.2566310325652x_{19} = 72.2566310325652
x20=36.1283155162826x_{20} = 36.1283155162826
x21=−36.1283155162826x_{21} = -36.1283155162826
x22=−48.6946861306418x_{22} = -48.6946861306418
x23=100.530964914873x_{23} = 100.530964914873
x24=−17.2787595947439x_{24} = -17.2787595947439
x25=−53.4070751110265x_{25} = -53.4070751110265
x26=−1.5707963267949x_{26} = -1.5707963267949
x27=51.8362787842316x_{27} = 51.8362787842316
x28=78.5398163397448x_{28} = 78.5398163397448
x29=12.5663706143592x_{29} = 12.5663706143592
x30=−61.261056745001x_{30} = -61.261056745001
x31=590.619418874881x_{31} = 590.619418874881
x32=−80.1106126665397x_{32} = -80.1106126665397
x33=−64.4026493985908x_{33} = -64.4026493985908
x34=67.5442420521806x_{34} = 67.5442420521806
x35=56.5486677646163x_{35} = 56.5486677646163
x36=86.3937979737193x_{36} = 86.3937979737193
x37=−73.8274273593601x_{37} = -73.8274273593601
x38=31.4159265358979x_{38} = 31.4159265358979
x39=28.2743338823081x_{39} = 28.2743338823081
x40=4.71238898038469x_{40} = 4.71238898038469
x41=−95.8185759344887x_{41} = -95.8185759344887
x42=−83.2522053201295x_{42} = -83.2522053201295
x43=−39.2699081698724x_{43} = -39.2699081698724
x44=65.9734457253857x_{44} = 65.9734457253857
x45=−14.1371669411541x_{45} = -14.1371669411541
x46=34.5575191894877x_{46} = 34.5575191894877
x47=14.1371669411541x_{47} = 14.1371669411541
x48=21.9911485751286x_{48} = 21.9911485751286
x49=−29.845130209103x_{49} = -29.845130209103
x50=−37.6991118430775x_{50} = -37.6991118430775
x51=−31.4159265358979x_{51} = -31.4159265358979
x52=−81.6814089933346x_{52} = -81.6814089933346
x53=48.6946861306418x_{53} = 48.6946861306418
x54=50.2654824574367x_{54} = 50.2654824574367
x55=45.553093477052x_{55} = 45.553093477052
x56=15.707963267949x_{56} = 15.707963267949
x57=−58.1194640914112x_{57} = -58.1194640914112
x58=58.1194640914112x_{58} = 58.1194640914112
x59=1.5707963267949x_{59} = 1.5707963267949
x60=7.85398163397448x_{60} = 7.85398163397448
x61=59.6902604182061x_{61} = 59.6902604182061
x62=43.9822971502571x_{62} = 43.9822971502571
x63=64.4026493985908x_{63} = 64.4026493985908
x64=20.4203522483337x_{64} = 20.4203522483337
x65=−65.9734457253857x_{65} = -65.9734457253857
x66=95.8185759344887x_{66} = 95.8185759344887
x67=0x_{67} = 0
x68=−42.4115008234622x_{68} = -42.4115008234622
x69=−483.805268652828x_{69} = -483.805268652828
x70=−59.6902604182061x_{70} = -59.6902604182061
x71=37.6991118430775x_{71} = 37.6991118430775
x72=29.845130209103x_{72} = 29.845130209103
x73=94.2477796076938x_{73} = 94.2477796076938
x74=6.28318530717959x_{74} = 6.28318530717959
x75=23.5619449019235x_{75} = 23.5619449019235
x76=−9.42477796076938x_{76} = -9.42477796076938
x77=−87.9645943005142x_{77} = -87.9645943005142
x78=73.8274273593601x_{78} = 73.8274273593601
x79=−28.2743338823081x_{79} = -28.2743338823081
x80=−94.2477796076938x_{80} = -94.2477796076938
x81=−7.85398163397448x_{81} = -7.85398163397448
x82=92.6769832808989x_{82} = 92.6769832808989
x83=−89.5353906273091x_{83} = -89.5353906273091
x84=−97.3893722612836x_{84} = -97.3893722612836
x85=−20.4203522483337x_{85} = -20.4203522483337
x86=−67.5442420521806x_{86} = -67.5442420521806
x87=42.4115008234622x_{87} = 42.4115008234622
x88=26.7035375555132x_{88} = 26.7035375555132
x89=89.5353906273091x_{89} = 89.5353906273091
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(2*x).
2sin⁡(2⋅0)2 \sin{\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
4cos⁡(2x)=04 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
The values of the extrema at the points:
 pi    
(--, 2)
 4     

 3*pi     
(----, -2)
  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=3π4x_{1} = \frac{3 \pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Increasing at intervals
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\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
−8sin⁡(2x)=0- 8 \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}

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]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
[0,π2]\left[0, \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(2 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(2sin⁡(2x))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(2 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\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:
2sin⁡(2x)=−2sin⁡(2x)2 \sin{\left(2 x \right)} = - 2 \sin{\left(2 x \right)}
- No
2sin⁡(2x)=2sin⁡(2x)2 \sin{\left(2 x \right)} = 2 \sin{\left(2 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = 2*sin(2*x)
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