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sin2x
Plotting a function graph/ sin2x

Graphing y = sin2x

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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
05-20-15-10-51015202530352-2
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=37.6991118430775x_{1} = 37.6991118430775
x2=26.7035375555132x_{2} = 26.7035375555132
x3=−21.9911485751286x_{3} = -21.9911485751286
x4=87.9645943005142x_{4} = 87.9645943005142
x5=78.5398163397448x_{5} = 78.5398163397448
x6=31.4159265358979x_{6} = 31.4159265358979
x7=70.6858347057703x_{7} = 70.6858347057703
x8=−42.4115008234622x_{8} = -42.4115008234622
x9=−97.3893722612836x_{9} = -97.3893722612836
x10=81.6814089933346x_{10} = 81.6814089933346
x11=−483.805268652828x_{11} = -483.805268652828
x12=21.9911485751286x_{12} = 21.9911485751286
x13=72.2566310325652x_{13} = 72.2566310325652
x14=95.8185759344887x_{14} = 95.8185759344887
x15=−29.845130209103x_{15} = -29.845130209103
x16=−86.3937979737193x_{16} = -86.3937979737193
x17=80.1106126665397x_{17} = 80.1106126665397
x18=−15.707963267949x_{18} = -15.707963267949
x19=−1.5707963267949x_{19} = -1.5707963267949
x20=−94.2477796076938x_{20} = -94.2477796076938
x21=−73.8274273593601x_{21} = -73.8274273593601
x22=−81.6814089933346x_{22} = -81.6814089933346
x23=−72.2566310325652x_{23} = -72.2566310325652
x24=56.5486677646163x_{24} = 56.5486677646163
x25=0x_{25} = 0
x26=59.6902604182061x_{26} = 59.6902604182061
x27=−7.85398163397448x_{27} = -7.85398163397448
x28=100.530964914873x_{28} = 100.530964914873
x29=−119.380520836412x_{29} = -119.380520836412
x30=45.553093477052x_{30} = 45.553093477052
x31=73.8274273593601x_{31} = 73.8274273593601
x32=6.28318530717959x_{32} = 6.28318530717959
x33=−37.6991118430775x_{33} = -37.6991118430775
x34=20.4203522483337x_{34} = 20.4203522483337
x35=−61.261056745001x_{35} = -61.261056745001
x36=65.9734457253857x_{36} = 65.9734457253857
x37=590.619418874881x_{37} = 590.619418874881
x38=−14.1371669411541x_{38} = -14.1371669411541
x39=−59.6902604182061x_{39} = -59.6902604182061
x40=29.845130209103x_{40} = 29.845130209103
x41=23.5619449019235x_{41} = 23.5619449019235
x42=−53.4070751110265x_{42} = -53.4070751110265
x43=86.3937979737193x_{43} = 86.3937979737193
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=14.1371669411541x_{45} = 14.1371669411541
x46=−40.8407044966673x_{46} = -40.8407044966673
x47=−43.9822971502571x_{47} = -43.9822971502571
x48=92.6769832808989x_{48} = 92.6769832808989
x49=1.5707963267949x_{49} = 1.5707963267949
x50=7.85398163397448x_{50} = 7.85398163397448
x51=−45.553093477052x_{51} = -45.553093477052
x52=−36.1283155162826x_{52} = -36.1283155162826
x53=64.4026493985908x_{53} = 64.4026493985908
x54=−95.8185759344887x_{54} = -95.8185759344887
x55=34.5575191894877x_{55} = 34.5575191894877
x56=67.5442420521806x_{56} = 67.5442420521806
x57=−48.6946861306418x_{57} = -48.6946861306418
x58=4.71238898038469x_{58} = 4.71238898038469
x59=36.1283155162826x_{59} = 36.1283155162826
x60=−51.8362787842316x_{60} = -51.8362787842316
x61=−39.2699081698724x_{61} = -39.2699081698724
x62=94.2477796076938x_{62} = 94.2477796076938
x63=−80.1106126665397x_{63} = -80.1106126665397
x64=−6.28318530717959x_{64} = -6.28318530717959
x65=12.5663706143592x_{65} = 12.5663706143592
x66=28.2743338823081x_{66} = 28.2743338823081
x67=−17.2787595947439x_{67} = -17.2787595947439
x68=−50.2654824574367x_{68} = -50.2654824574367
x69=113.097335529233x_{69} = 113.097335529233
x70=48.6946861306418x_{70} = 48.6946861306418
x71=−58.1194640914112x_{71} = -58.1194640914112
x72=−67.5442420521806x_{72} = -67.5442420521806
x73=15.707963267949x_{73} = 15.707963267949
x74=−20.4203522483337x_{74} = -20.4203522483337
x75=89.5353906273091x_{75} = 89.5353906273091
x76=−31.4159265358979x_{76} = -31.4159265358979
x77=−9.42477796076938x_{77} = -9.42477796076938
x78=−89.5353906273091x_{78} = -89.5353906273091
x79=−75.398223686155x_{79} = -75.398223686155
x80=−28.2743338823081x_{80} = -28.2743338823081
x81=−64.4026493985908x_{81} = -64.4026493985908
x82=−87.9645943005142x_{82} = -87.9645943005142
x83=42.4115008234622x_{83} = 42.4115008234622
x84=43.9822971502571x_{84} = 43.9822971502571
x85=58.1194640914112x_{85} = 58.1194640914112
x86=50.2654824574367x_{86} = 50.2654824574367
x87=51.8362787842316x_{87} = 51.8362787842316
x88=−83.2522053201295x_{88} = -83.2522053201295
x89=−23.5619449019235x_{89} = -23.5619449019235
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⁡(2⋅0)\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
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
The graph
Graphing y = sin2x
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