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

Graphing y = sin(2*x)/2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       sin(2*x)
f(x) = --------
          2
f(x)=sin⁡(2x)2f{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{2} 
f = sin(2*x)/2
The graph of the function
02468-8-6-4-2-10101-1
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)2=0\frac{\sin{\left(2 x \right)}}{2} = 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=89.5353906273091x_{1} = 89.5353906273091
x2=−15.707963267949x_{2} = -15.707963267949
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=42.4115008234622x_{4} = 42.4115008234622
x5=21.9911485751286x_{5} = 21.9911485751286
x6=0x_{6} = 0
x7=−42.4115008234622x_{7} = -42.4115008234622
x8=−29.845130209103x_{8} = -29.845130209103
x9=−21.9911485751286x_{9} = -21.9911485751286
x10=−36.1283155162826x_{10} = -36.1283155162826
x11=28.2743338823081x_{11} = 28.2743338823081
x12=86.3937979737193x_{12} = 86.3937979737193
x13=72.2566310325652x_{13} = 72.2566310325652
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=−61.261056745001x_{15} = -61.261056745001
x16=−40.8407044966673x_{16} = -40.8407044966673
x17=87.9645943005142x_{17} = 87.9645943005142
x18=−95.8185759344887x_{18} = -95.8185759344887
x19=−50.2654824574367x_{19} = -50.2654824574367
x20=23.5619449019235x_{20} = 23.5619449019235
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=−97.3893722612836x_{22} = -97.3893722612836
x23=50.2654824574367x_{23} = 50.2654824574367
x24=590.619418874881x_{24} = 590.619418874881
x25=−14.1371669411541x_{25} = -14.1371669411541
x26=59.6902604182061x_{26} = 59.6902604182061
x27=−119.380520836412x_{27} = -119.380520836412
x28=58.1194640914112x_{28} = 58.1194640914112
x29=−53.4070751110265x_{29} = -53.4070751110265
x30=−48.6946861306418x_{30} = -48.6946861306418
x31=−23.5619449019235x_{31} = -23.5619449019235
x32=−86.3937979737193x_{32} = -86.3937979737193
x33=−17.2787595947439x_{33} = -17.2787595947439
x34=12.5663706143592x_{34} = 12.5663706143592
x35=−81.6814089933346x_{35} = -81.6814089933346
x36=94.2477796076938x_{36} = 94.2477796076938
x37=81.6814089933346x_{37} = 81.6814089933346
x38=−67.5442420521806x_{38} = -67.5442420521806
x39=−80.1106126665397x_{39} = -80.1106126665397
x40=−1.5707963267949x_{40} = -1.5707963267949
x41=31.4159265358979x_{41} = 31.4159265358979
x42=92.6769832808989x_{42} = 92.6769832808989
x43=36.1283155162826x_{43} = 36.1283155162826
x44=−39.2699081698724x_{44} = -39.2699081698724
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=4.71238898038469x_{46} = 4.71238898038469
x47=48.6946861306418x_{47} = 48.6946861306418
x48=−72.2566310325652x_{48} = -72.2566310325652
x49=37.6991118430775x_{49} = 37.6991118430775
x50=70.6858347057703x_{50} = 70.6858347057703
x51=−45.553093477052x_{51} = -45.553093477052
x52=−89.5353906273091x_{52} = -89.5353906273091
x53=65.9734457253857x_{53} = 65.9734457253857
x54=73.8274273593601x_{54} = 73.8274273593601
x55=20.4203522483337x_{55} = 20.4203522483337
x56=−483.805268652828x_{56} = -483.805268652828
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=1.5707963267949x_{58} = 1.5707963267949
x59=45.553093477052x_{59} = 45.553093477052
x60=78.5398163397448x_{60} = 78.5398163397448
x61=−6.28318530717959x_{61} = -6.28318530717959
x62=95.8185759344887x_{62} = 95.8185759344887
x63=15.707963267949x_{63} = 15.707963267949
x64=−20.4203522483337x_{64} = -20.4203522483337
x65=−58.1194640914112x_{65} = -58.1194640914112
x66=56.5486677646163x_{66} = 56.5486677646163
x67=113.097335529233x_{67} = 113.097335529233
x68=80.1106126665397x_{68} = 80.1106126665397
x69=7.85398163397448x_{69} = 7.85398163397448
x70=26.7035375555132x_{70} = 26.7035375555132
x71=29.845130209103x_{71} = 29.845130209103
x72=−65.9734457253857x_{72} = -65.9734457253857
x73=43.9822971502571x_{73} = 43.9822971502571
x74=14.1371669411541x_{74} = 14.1371669411541
x75=−37.6991118430775x_{75} = -37.6991118430775
x76=−59.6902604182061x_{76} = -59.6902604182061
x77=−64.4026493985908x_{77} = -64.4026493985908
x78=−83.2522053201295x_{78} = -83.2522053201295
x79=−75.398223686155x_{79} = -75.398223686155
x80=51.8362787842316x_{80} = 51.8362787842316
x81=100.530964914873x_{81} = 100.530964914873
x82=64.4026493985908x_{82} = 64.4026493985908
x83=34.5575191894877x_{83} = 34.5575191894877
x84=−73.8274273593601x_{84} = -73.8274273593601
x85=6.28318530717959x_{85} = 6.28318530717959
x86=−9.42477796076938x_{86} = -9.42477796076938
x87=−51.8362787842316x_{87} = -51.8362787842316
x88=67.5442420521806x_{88} = 67.5442420521806
x89=−7.85398163397448x_{89} = -7.85398163397448
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⁡(0⋅2)2\frac{\sin{\left(0 \cdot 2 \right)}}{2}
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
cos⁡(2x)=0\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   2 

 3*pi       
(----, -1/2)
  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
−2sin⁡(2x)=0- 2 \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)2)=⟨−12,12⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
lim⁡x→∞(sin⁡(2x)2)=⟨−12,12⟩\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\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⁡(2x)2x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x)2x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{2 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)2=−sin⁡(2x)2\frac{\sin{\left(2 x \right)}}{2} = - \frac{\sin{\left(2 x \right)}}{2}
- No
sin⁡(2x)2=sin⁡(2x)2\frac{\sin{\left(2 x \right)}}{2} = \frac{\sin{\left(2 x \right)}}{2}
- No
so, the function
not is
neither even, nor odd
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