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Plotting a function graph/ sin(2*x)

Graphing y = sin(2*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(2*x)
f(x)=sin⁡(2x)f{\left(x \right)} = \sin{\left(2 x \right)} 
f = sin(2*x)
The graph of the function
0.003.000.250.500.751.001.251.501.752.002.252.502.752-2
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)=0\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=31.4159265358979x_{1} = 31.4159265358979
x2=86.3937979737193x_{2} = 86.3937979737193
x3=92.6769832808989x_{3} = 92.6769832808989
x4=−7.85398163397448x_{4} = -7.85398163397448
x5=1.5707963267949x_{5} = 1.5707963267949
x6=−64.4026493985908x_{6} = -64.4026493985908
x7=−58.1194640914112x_{7} = -58.1194640914112
x8=−20.4203522483337x_{8} = -20.4203522483337
x9=−45.553093477052x_{9} = -45.553093477052
x10=64.4026493985908x_{10} = 64.4026493985908
x11=−483.805268652828x_{11} = -483.805268652828
x12=−29.845130209103x_{12} = -29.845130209103
x13=−9.42477796076938x_{13} = -9.42477796076938
x14=70.6858347057703x_{14} = 70.6858347057703
x15=42.4115008234622x_{15} = 42.4115008234622
x16=43.9822971502571x_{16} = 43.9822971502571
x17=−31.4159265358979x_{17} = -31.4159265358979
x18=78.5398163397448x_{18} = 78.5398163397448
x19=−61.261056745001x_{19} = -61.261056745001
x20=−15.707963267949x_{20} = -15.707963267949
x21=−28.2743338823081x_{21} = -28.2743338823081
x22=−21.9911485751286x_{22} = -21.9911485751286
x23=95.8185759344887x_{23} = 95.8185759344887
x24=56.5486677646163x_{24} = 56.5486677646163
x25=58.1194640914112x_{25} = 58.1194640914112
x26=−73.8274273593601x_{26} = -73.8274273593601
x27=7.85398163397448x_{27} = 7.85398163397448
x28=−72.2566310325652x_{28} = -72.2566310325652
x29=0x_{29} = 0
x30=−50.2654824574367x_{30} = -50.2654824574367
x31=113.097335529233x_{31} = 113.097335529233
x32=89.5353906273091x_{32} = 89.5353906273091
x33=23.5619449019235x_{33} = 23.5619449019235
x34=−48.6946861306418x_{34} = -48.6946861306418
x35=−42.4115008234622x_{35} = -42.4115008234622
x36=−67.5442420521806x_{36} = -67.5442420521806
x37=−94.2477796076938x_{37} = -94.2477796076938
x38=20.4203522483337x_{38} = 20.4203522483337
x39=12.5663706143592x_{39} = 12.5663706143592
x40=−59.6902604182061x_{40} = -59.6902604182061
x41=−14.1371669411541x_{41} = -14.1371669411541
x42=−119.380520836412x_{42} = -119.380520836412
x43=29.845130209103x_{43} = 29.845130209103
x44=100.530964914873x_{44} = 100.530964914873
x45=−97.3893722612836x_{45} = -97.3893722612836
x46=26.7035375555132x_{46} = 26.7035375555132
x47=48.6946861306418x_{47} = 48.6946861306418
x48=−86.3937979737193x_{48} = -86.3937979737193
x49=80.1106126665397x_{49} = 80.1106126665397
x50=21.9911485751286x_{50} = 21.9911485751286
x51=−40.8407044966673x_{51} = -40.8407044966673
x52=−95.8185759344887x_{52} = -95.8185759344887
x53=−36.1283155162826x_{53} = -36.1283155162826
x54=34.5575191894877x_{54} = 34.5575191894877
x55=51.8362787842316x_{55} = 51.8362787842316
x56=−81.6814089933346x_{56} = -81.6814089933346
x57=50.2654824574367x_{57} = 50.2654824574367
x58=−87.9645943005142x_{58} = -87.9645943005142
x59=37.6991118430775x_{59} = 37.6991118430775
x60=−1.5707963267949x_{60} = -1.5707963267949
x61=590.619418874881x_{61} = 590.619418874881
x62=28.2743338823081x_{62} = 28.2743338823081
x63=−65.9734457253857x_{63} = -65.9734457253857
x64=59.6902604182061x_{64} = 59.6902604182061
x65=−83.2522053201295x_{65} = -83.2522053201295
x66=72.2566310325652x_{66} = 72.2566310325652
x67=−17.2787595947439x_{67} = -17.2787595947439
x68=45.553093477052x_{68} = 45.553093477052
x69=−6.28318530717959x_{69} = -6.28318530717959
x70=6.28318530717959x_{70} = 6.28318530717959
x71=−51.8362787842316x_{71} = -51.8362787842316
x72=94.2477796076938x_{72} = 94.2477796076938
x73=−37.6991118430775x_{73} = -37.6991118430775
x74=−39.2699081698724x_{74} = -39.2699081698724
x75=65.9734457253857x_{75} = 65.9734457253857
x76=−43.9822971502571x_{76} = -43.9822971502571
x77=4.71238898038469x_{77} = 4.71238898038469
x78=87.9645943005142x_{78} = 87.9645943005142
x79=36.1283155162826x_{79} = 36.1283155162826
x80=−53.4070751110265x_{80} = -53.4070751110265
x81=−23.5619449019235x_{81} = -23.5619449019235
x82=73.8274273593601x_{82} = 73.8274273593601
x83=81.6814089933346x_{83} = 81.6814089933346
x84=14.1371669411541x_{84} = 14.1371669411541
x85=−75.398223686155x_{85} = -75.398223686155
x86=−89.5353906273091x_{86} = -89.5353906273091
x87=15.707963267949x_{87} = 15.707963267949
x88=−80.1106126665397x_{88} = -80.1106126665397
x89=67.5442420521806x_{89} = 67.5442420521806
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).
sin⁡(0⋅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
2cos⁡(2x)=02 \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    
(--, 1)
 4     

 3*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=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
−4sin⁡(2x)=0- 4 \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→−∞sin⁡(2x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(2 x \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)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(2 x \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), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\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→∞(sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\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:
sin⁡(2x)=−sin⁡(2x)\sin{\left(2 x \right)} = - \sin{\left(2 x \right)}
- No
sin⁡(2x)=sin⁡(2x)\sin{\left(2 x \right)} = \sin{\left(2 x \right)}
- Yes
so, the function
is
odd
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