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Plotting a function graph/ sin2x+sinx

Graphing y = sin2x+sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(2*x) + sin(x)
f(x)=sin⁡(x)+sin⁡(2x)f{\left(x \right)} = \sin{\left(x \right)} + \sin{\left(2 x \right)} 
f = sin(x) + 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:
sin⁡(x)+sin⁡(2x)=0\sin{\left(x \right)} + \sin{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−πx_{3} = - \pi
x4=−2π3x_{4} = - \frac{2 \pi}{3}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
x7=4π3x_{7} = \frac{4 \pi}{3}
x8=2πx_{8} = 2 \pi
Numerical solution
x1=62.8318530717959x_{1} = 62.8318530717959
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=85.870199198121x_{3} = 85.870199198121
x4=−23.0383461263252x_{4} = -23.0383461263252
x5=73.3038285837618x_{5} = 73.3038285837618
x6=−41.8879020478639x_{6} = -41.8879020478639
x7=79.5870138909414x_{7} = 79.5870138909414
x8=−77.4926187885482x_{8} = -77.4926187885482
x9=72.2566310325652x_{9} = 72.2566310325652
x10=−43.9822971502571x_{10} = -43.9822971502571
x11=−90.0589894029074x_{11} = -90.0589894029074
x12=−69.1150383789755x_{12} = -69.1150383789755
x13=−37.6991118430775x_{13} = -37.6991118430775
x14=34.5575191894877x_{14} = 34.5575191894877
x15=−12.5663706143592x_{15} = -12.5663706143592
x16=−94.2477796076938x_{16} = -94.2477796076938
x17=52.3598775598299x_{17} = 52.3598775598299
x18=−48.1710873550435x_{18} = -48.1710873550435
x19=−14.6607657167524x_{19} = -14.6607657167524
x20=−56.5486677646163x_{20} = -56.5486677646163
x21=−21.9911485751286x_{21} = -21.9911485751286
x22=50.2654824574367x_{22} = 50.2654824574367
x23=−15.707963267949x_{23} = -15.707963267949
x24=−46.0766922526503x_{24} = -46.0766922526503
x25=−2.0943951023932x_{25} = -2.0943951023932
x26=−5612.97887441376x_{26} = -5612.97887441376
x27=21.9911485751286x_{27} = 21.9911485751286
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=−73.3038285837618x_{29} = -73.3038285837618
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=20.943951023932x_{31} = 20.943951023932
x32=−92.1533845053006x_{32} = -92.1533845053006
x33=−52.3598775598299x_{33} = -52.3598775598299
x34=−67.0206432765823x_{34} = -67.0206432765823
x35=4.18879020478639x_{35} = 4.18879020478639
x36=59.6902604182061x_{36} = 59.6902604182061
x37=41.8879020478639x_{37} = 41.8879020478639
x38=−9.42477796076938x_{38} = -9.42477796076938
x39=−85.870199198121x_{39} = -85.870199198121
x40=31.4159265358979x_{40} = 31.4159265358979
x41=35.6047167406843x_{41} = 35.6047167406843
x42=48.1710873550435x_{42} = 48.1710873550435
x43=10.471975511966x_{43} = 10.471975511966
x44=−35.6047167406843x_{44} = -35.6047167406843
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=46.0766922526503x_{46} = 46.0766922526503
x47=15.707963267949x_{47} = 15.707963267949
x48=−83.7758040957278x_{48} = -83.7758040957278
x49=28.2743338823081x_{49} = 28.2743338823081
x50=94.2477796076938x_{50} = 94.2477796076938
x51=39.7935069454707x_{51} = 39.7935069454707
x52=−79.5870138909414x_{52} = -79.5870138909414
x53=37.6991118430775x_{53} = 37.6991118430775
x54=96.342174710087x_{54} = 96.342174710087
x55=6.28318530717959x_{55} = 6.28318530717959
x56=−53.4070751110265x_{56} = -53.4070751110265
x57=14.6607657167524x_{57} = 14.6607657167524
x58=−58.6430628670095x_{58} = -58.6430628670095
x59=83.7758040957278x_{59} = 83.7758040957278
x60=−6.28318530717959x_{60} = -6.28318530717959
x61=8.37758040957278x_{61} = 8.37758040957278
x62=−75.398223686155x_{62} = -75.398223686155
x63=−33.5103216382911x_{63} = -33.5103216382911
x64=−29.3215314335047x_{64} = -29.3215314335047
x65=64.9262481741891x_{65} = 64.9262481741891
x66=0x_{66} = 0
x67=−39.7935069454707x_{67} = -39.7935069454707
x68=2.0943951023932x_{68} = 2.0943951023932
x69=54.4542726622231x_{69} = 54.4542726622231
x70=−25.1327412287183x_{70} = -25.1327412287183
x71=−28.2743338823081x_{71} = -28.2743338823081
x72=81.6814089933346x_{72} = 81.6814089933346
x73=98.4365698124802x_{73} = 98.4365698124802
x74=100.530964914873x_{74} = 100.530964914873
x75=−8.37758040957278x_{75} = -8.37758040957278
x76=78.5398163397448x_{76} = 78.5398163397448
x77=29.3215314335047x_{77} = 29.3215314335047
x78=92.1533845053006x_{78} = 92.1533845053006
x79=12.5663706143592x_{79} = 12.5663706143592
x80=87.9645943005142x_{80} = 87.9645943005142
x81=−100.530964914873x_{81} = -100.530964914873
x82=58.6430628670095x_{82} = 58.6430628670095
x83=18.8495559215388x_{83} = 18.8495559215388
x84=43.9822971502571x_{84} = 43.9822971502571
x85=−31.4159265358979x_{85} = -31.4159265358979
x86=−81.6814089933346x_{86} = -81.6814089933346
x87=56.5486677646163x_{87} = 56.5486677646163
x88=−4.18879020478639x_{88} = -4.18879020478639
x89=−59.6902604182061x_{89} = -59.6902604182061
x90=65.9734457253857x_{90} = 65.9734457253857
x91=90.0589894029074x_{91} = 90.0589894029074
x92=−96.342174710087x_{92} = -96.342174710087
x93=75.398223686155x_{93} = 75.398223686155
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(x).
sin⁡(0⋅2)+sin⁡(0)\sin{\left(0 \cdot 2 \right)} + \sin{\left(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
cos⁡(x)+2cos⁡(2x)=0\cos{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−ilog⁡(−18+338−2i33+158)x_{1} = - i \log{\left(- \frac{1}{8} + \frac{\sqrt{33}}{8} - \frac{\sqrt{2} i \sqrt{\sqrt{33} + 15}}{8} \right)}
x2=−ilog⁡(−18+338+2i33+158)x_{2} = - i \log{\left(- \frac{1}{8} + \frac{\sqrt{33}}{8} + \frac{\sqrt{2} i \sqrt{\sqrt{33} + 15}}{8} \right)}
x3=−ilog⁡(−338−18−2i15−338)x_{3} = - i \log{\left(- \frac{\sqrt{33}}{8} - \frac{1}{8} - \frac{\sqrt{2} i \sqrt{15 - \sqrt{33}}}{8} \right)}
x4=−ilog⁡(−338−18+2i15−338)x_{4} = - i \log{\left(- \frac{\sqrt{33}}{8} - \frac{1}{8} + \frac{\sqrt{2} i \sqrt{15 - \sqrt{33}}}{8} \right)}
The values of the extrema at the points:
       /                          _____________\       /     /                          _____________\\      /       /                          _____________\\ 
       |        ____       ___   /        ____ |       |     |        ____       ___   /        ____ ||      |       |        ____       ___   /        ____ || 
       |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  |       |     |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  ||      |       |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  || 
(-I*log|- - + ------ - ------------------------|, - sin|I*log|- - + ------ - ------------------------|| - sin|2*I*log|- - + ------ - ------------------------||)
       \  8     8                 8            /       \     \  8     8                 8            //      \       \  8     8                 8            // 

       /                          _____________\       /     /                          _____________\\      /       /                          _____________\\ 
       |        ____       ___   /        ____ |       |     |        ____       ___   /        ____ ||      |       |        ____       ___   /        ____ || 
       |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  |       |     |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  ||      |       |  1   \/ 33    I*\/ 2 *\/  15 + \/ 33  || 
(-I*log|- - + ------ + ------------------------|, - sin|I*log|- - + ------ + ------------------------|| - sin|2*I*log|- - + ------ + ------------------------||)
       \  8     8                 8            /       \     \  8     8                 8            //      \       \  8     8                 8            // 

       /                          _____________\       /     /                          _____________\\      /       /                          _____________\\ 
       |        ____       ___   /        ____ |       |     |        ____       ___   /        ____ ||      |       |        ____       ___   /        ____ || 
       |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  |       |     |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  ||      |       |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  || 
(-I*log|- - - ------ - ------------------------|, - sin|I*log|- - - ------ - ------------------------|| - sin|2*I*log|- - - ------ - ------------------------||)
       \  8     8                 8            /       \     \  8     8                 8            //      \       \  8     8                 8            // 

       /                          _____________\       /     /                          _____________\\      /       /                          _____________\\ 
       |        ____       ___   /        ____ |       |     |        ____       ___   /        ____ ||      |       |        ____       ___   /        ____ || 
       |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  |       |     |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  ||      |       |  1   \/ 33    I*\/ 2 *\/  15 - \/ 33  || 
(-I*log|- - - ------ + ------------------------|, - sin|I*log|- - - ------ + ------------------------|| - sin|2*I*log|- - - ------ + ------------------------||)
       \  8     8                 8            /       \     \  8     8                 8            //      \       \  8     8                 8            //


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=−atan⁡(233+15−1+33)x_{1} = - \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{-1 + \sqrt{33}} \right)}
x2=atan⁡(215−33−33−1)+πx_{2} = \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{- \sqrt{33} - 1} \right)} + \pi
Maxima of the function at points:
x2=atan⁡(233+15−1+33)x_{2} = \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{-1 + \sqrt{33}} \right)}
x2=−π−atan⁡(215−33−33−1)x_{2} = - \pi - \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{- \sqrt{33} - 1} \right)}
Decreasing at intervals
[atan⁡(215−33−33−1)+π,∞)\left[\operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{- \sqrt{33} - 1} \right)} + \pi, \infty\right)
Increasing at intervals
(−∞,−atan⁡(233+15−1+33)]∪[atan⁡(233+15−1+33),atan⁡(215−33−33−1)+π]\left(-\infty, - \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{-1 + \sqrt{33}} \right)}\right] \cup \left[\operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{-1 + \sqrt{33}} \right)}, \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{- \sqrt{33} - 1} \right)} + \pi\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
−(sin⁡(x)+4sin⁡(2x))=0- (\sin{\left(x \right)} + 4 \sin{\left(2 x \right)}) = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=−ilog⁡(−18−37i8)x_{3} = - i \log{\left(- \frac{1}{8} - \frac{3 \sqrt{7} i}{8} \right)}
x4=−ilog⁡(−18+37i8)x_{4} = - i \log{\left(- \frac{1}{8} + \frac{3 \sqrt{7} i}{8} \right)}

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
[−π+atan⁡(37),0]∪[π−atan⁡(37),∞)\left[- \pi + \operatorname{atan}{\left(3 \sqrt{7} \right)}, 0\right] \cup \left[\pi - \operatorname{atan}{\left(3 \sqrt{7} \right)}, \infty\right)
Convex at the intervals
(−∞,−π+atan⁡(37)]\left(-\infty, - \pi + \operatorname{atan}{\left(3 \sqrt{7} \right)}\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⁡(x)+sin⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \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→∞(sin⁡(x)+sin⁡(2x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} + \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 sin(2*x) + sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)+sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \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⁡(x)+sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \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⁡(x)+sin⁡(2x)=−sin⁡(x)−sin⁡(2x)\sin{\left(x \right)} + \sin{\left(2 x \right)} = - \sin{\left(x \right)} - \sin{\left(2 x \right)}
- No
sin⁡(x)+sin⁡(2x)=sin⁡(x)+sin⁡(2x)\sin{\left(x \right)} + \sin{\left(2 x \right)} = \sin{\left(x \right)} + \sin{\left(2 x \right)}
- No
so, the function
not is
neither even, nor odd
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