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Plotting a function graph/ 2sin(2x)

Graphing y = 2sin(2x)

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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
02468-8-6-4-2-10105-5
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=−86.3937979737193x_{1} = -86.3937979737193
x2=29.845130209103x_{2} = 29.845130209103
x3=14.1371669411541x_{3} = 14.1371669411541
x4=48.6946861306418x_{4} = 48.6946861306418
x5=−72.2566310325652x_{5} = -72.2566310325652
x6=−80.1106126665397x_{6} = -80.1106126665397
x7=−7.85398163397448x_{7} = -7.85398163397448
x8=−59.6902604182061x_{8} = -59.6902604182061
x9=−6.28318530717959x_{9} = -6.28318530717959
x10=12.5663706143592x_{10} = 12.5663706143592
x11=−119.380520836412x_{11} = -119.380520836412
x12=67.5442420521806x_{12} = 67.5442420521806
x13=56.5486677646163x_{13} = 56.5486677646163
x14=−58.1194640914112x_{14} = -58.1194640914112
x15=95.8185759344887x_{15} = 95.8185759344887
x16=72.2566310325652x_{16} = 72.2566310325652
x17=37.6991118430775x_{17} = 37.6991118430775
x18=80.1106126665397x_{18} = 80.1106126665397
x19=100.530964914873x_{19} = 100.530964914873
x20=94.2477796076938x_{20} = 94.2477796076938
x21=−14.1371669411541x_{21} = -14.1371669411541
x22=7.85398163397448x_{22} = 7.85398163397448
x23=−29.845130209103x_{23} = -29.845130209103
x24=70.6858347057703x_{24} = 70.6858347057703
x25=−20.4203522483337x_{25} = -20.4203522483337
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=−483.805268652828x_{27} = -483.805268652828
x28=−31.4159265358979x_{28} = -31.4159265358979
x29=59.6902604182061x_{29} = 59.6902604182061
x30=42.4115008234622x_{30} = 42.4115008234622
x31=0x_{31} = 0
x32=−17.2787595947439x_{32} = -17.2787595947439
x33=−67.5442420521806x_{33} = -67.5442420521806
x34=113.097335529233x_{34} = 113.097335529233
x35=50.2654824574367x_{35} = 50.2654824574367
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=45.553093477052x_{37} = 45.553093477052
x38=−45.553093477052x_{38} = -45.553093477052
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=−23.5619449019235x_{40} = -23.5619449019235
x41=590.619418874881x_{41} = 590.619418874881
x42=58.1194640914112x_{42} = 58.1194640914112
x43=−36.1283155162826x_{43} = -36.1283155162826
x44=87.9645943005142x_{44} = 87.9645943005142
x45=−51.8362787842316x_{45} = -51.8362787842316
x46=−73.8274273593601x_{46} = -73.8274273593601
x47=4.71238898038469x_{47} = 4.71238898038469
x48=64.4026493985908x_{48} = 64.4026493985908
x49=−42.4115008234622x_{49} = -42.4115008234622
x50=−95.8185759344887x_{50} = -95.8185759344887
x51=−75.398223686155x_{51} = -75.398223686155
x52=81.6814089933346x_{52} = 81.6814089933346
x53=−65.9734457253857x_{53} = -65.9734457253857
x54=−37.6991118430775x_{54} = -37.6991118430775
x55=1.5707963267949x_{55} = 1.5707963267949
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=20.4203522483337x_{57} = 20.4203522483337
x58=23.5619449019235x_{58} = 23.5619449019235
x59=−1.5707963267949x_{59} = -1.5707963267949
x60=92.6769832808989x_{60} = 92.6769832808989
x61=6.28318530717959x_{61} = 6.28318530717959
x62=28.2743338823081x_{62} = 28.2743338823081
x63=−83.2522053201295x_{63} = -83.2522053201295
x64=−94.2477796076938x_{64} = -94.2477796076938
x65=86.3937979737193x_{65} = 86.3937979737193
x66=43.9822971502571x_{66} = 43.9822971502571
x67=−9.42477796076938x_{67} = -9.42477796076938
x68=65.9734457253857x_{68} = 65.9734457253857
x69=31.4159265358979x_{69} = 31.4159265358979
x70=89.5353906273091x_{70} = 89.5353906273091
x71=−15.707963267949x_{71} = -15.707963267949
x72=−64.4026493985908x_{72} = -64.4026493985908
x73=−50.2654824574367x_{73} = -50.2654824574367
x74=−48.6946861306418x_{74} = -48.6946861306418
x75=36.1283155162826x_{75} = 36.1283155162826
x76=15.707963267949x_{76} = 15.707963267949
x77=51.8362787842316x_{77} = 51.8362787842316
x78=26.7035375555132x_{78} = 26.7035375555132
x79=73.8274273593601x_{79} = 73.8274273593601
x80=−39.2699081698724x_{80} = -39.2699081698724
x81=21.9911485751286x_{81} = 21.9911485751286
x82=34.5575191894877x_{82} = 34.5575191894877
x83=−97.3893722612836x_{83} = -97.3893722612836
x84=−40.8407044966673x_{84} = -40.8407044966673
x85=−28.2743338823081x_{85} = -28.2743338823081
x86=78.5398163397448x_{86} = 78.5398163397448
x87=−89.5353906273091x_{87} = -89.5353906273091
x88=−81.6814089933346x_{88} = -81.6814089933346
x89=−61.261056745001x_{89} = -61.261056745001
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⁡(0⋅2)2 \sin{\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
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
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