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Plotting a function graph/ sinx+(1/3)sin3x

Graphing y = sinx+(1/3)sin3x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                sin(3*x)
f(x) = sin(x) + --------
                   3
f(x)=sin⁡(x)+sin⁡(3x)3f{\left(x \right)} = \sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3} 
f = sin(x) + sin(3*x)/3
The graph of the function
02468-8-6-4-2-10102-2
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⁡(x)+sin⁡(3x)3=0\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−43.9822971502571x_{1} = -43.9822971502571
x2=−31.4159265358979x_{2} = -31.4159265358979
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=91.106186954104x_{4} = 91.106186954104
x5=6.28318530717959x_{5} = 6.28318530717959
x6=−72.2566310325652x_{6} = -72.2566310325652
x7=94.2477796076938x_{7} = 94.2477796076938
x8=50.2654824574367x_{8} = 50.2654824574367
x9=56.5486677646163x_{9} = 56.5486677646163
x10=47.1238898038469x_{10} = 47.1238898038469
x11=43.9822971502571x_{11} = 43.9822971502571
x12=−50.2654824574367x_{12} = -50.2654824574367
x13=37.6991118430775x_{13} = 37.6991118430775
x14=−28.2743338823081x_{14} = -28.2743338823081
x15=65.9734457253857x_{15} = 65.9734457253857
x16=15.707963267949x_{16} = 15.707963267949
x17=28.2743338823081x_{17} = 28.2743338823081
x18=−40.8407044966673x_{18} = -40.8407044966673
x19=−6.28318530717959x_{19} = -6.28318530717959
x20=−81.6814089933346x_{20} = -81.6814089933346
x21=−15.707963267949x_{21} = -15.707963267949
x22=−59.6902604182061x_{22} = -59.6902604182061
x23=72.2566310325652x_{23} = 72.2566310325652
x24=−25.1327412287183x_{24} = -25.1327412287183
x25=21.9911485751286x_{25} = 21.9911485751286
x26=−75.398223686155x_{26} = -75.398223686155
x27=−103.672557568463x_{27} = -103.672557568463
x28=78.5398163397448x_{28} = 78.5398163397448
x29=−53.4070751110265x_{29} = -53.4070751110265
x30=−2544.69004940773x_{30} = -2544.69004940773
x31=−380.132711084365x_{31} = -380.132711084365
x32=25.1327412287183x_{32} = 25.1327412287183
x33=100.530964914873x_{33} = 100.530964914873
x34=−87.9645943005142x_{34} = -87.9645943005142
x35=−9.42477796076938x_{35} = -9.42477796076938
x36=254.469004940773x_{36} = 254.469004940773
x37=81.6814089933346x_{37} = 81.6814089933346
x38=87.9645943005142x_{38} = 87.9645943005142
x39=12.5663706143592x_{39} = 12.5663706143592
x40=0x_{40} = 0
x41=−21.9911485751286x_{41} = -21.9911485751286
x42=−37.6991118430775x_{42} = -37.6991118430775
x43=−78.5398163397448x_{43} = -78.5398163397448
x44=−94.2477796076938x_{44} = -94.2477796076938
x45=59.6902604182061x_{45} = 59.6902604182061
x46=34.5575191894877x_{46} = 34.5575191894877
x47=−65.9734457253857x_{47} = -65.9734457253857
x48=18.8495559215388x_{48} = 18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) + sin(3*x)/3.
sin⁡(0)+sin⁡(0⋅3)3\sin{\left(0 \right)} + \frac{\sin{\left(0 \cdot 3 \right)}}{3}
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)+cos⁡(3x)=0\cos{\left(x \right)} + \cos{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=−3π4x_{1} = - \frac{3 \pi}{4}
x2=−π2x_{2} = - \frac{\pi}{2}
x3=−π4x_{3} = - \frac{\pi}{4}
x4=π4x_{4} = \frac{\pi}{4}
x5=π2x_{5} = \frac{\pi}{2}
x6=3π4x_{6} = \frac{3 \pi}{4}
The values of the extrema at the points:
             ___ 
 -3*pi  -2*\/ 2  
(-----, --------)
   4       3     

 -pi        
(----, -2/3)
  2         

            ___ 
 -pi   -2*\/ 2  
(----, --------)
  4       3     

         ___ 
 pi  2*\/ 2  
(--, -------)
 4      3    

 pi      
(--, 2/3)
 2       

           ___ 
 3*pi  2*\/ 2  
(----, -------)
  4       3


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}
x2=−π4x_{2} = - \frac{\pi}{4}
x3=π2x_{3} = \frac{\pi}{2}
Maxima of the function at points:
x3=−π2x_{3} = - \frac{\pi}{2}
x3=π4x_{3} = \frac{\pi}{4}
x3=3π4x_{3} = \frac{3 \pi}{4}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−3π4]\left(-\infty, - \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
−(sin⁡(x)+3sin⁡(3x))=0- (\sin{\left(x \right)} + 3 \sin{\left(3 x \right)}) = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=i(log⁡(3)−log⁡(−2−5i))2x_{3} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 - \sqrt{5} i \right)}\right)}{2}
x4=i(log⁡(3)−log⁡(−2+5i))2x_{4} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 + \sqrt{5} i \right)}\right)}{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
[π,∞)\left[\pi, \infty\right)
Convex at the intervals
(−∞,−π2+atan⁡(52)2]∪[0,−atan⁡(52)2+π2]\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2}\right] \cup \left[0, - \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} + \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⁡(x)+sin⁡(3x)3)=⟨−43,43⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}\right) = \left\langle - \frac{4}{3}, \frac{4}{3}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−43,43⟩y = \left\langle - \frac{4}{3}, \frac{4}{3}\right\rangle
lim⁡x→∞(sin⁡(x)+sin⁡(3x)3)=⟨−43,43⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}\right) = \left\langle - \frac{4}{3}, \frac{4}{3}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−43,43⟩y = \left\langle - \frac{4}{3}, \frac{4}{3}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x) + sin(3*x)/3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)+sin⁡(3x)3x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)+sin⁡(3x)3x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}}{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⁡(3x)3=−sin⁡(x)−sin⁡(3x)3\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3} = - \sin{\left(x \right)} - \frac{\sin{\left(3 x \right)}}{3}
- No
sin⁡(x)+sin⁡(3x)3=sin⁡(x)+sin⁡(3x)3\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3} = \sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}
- No
so, the function
not is
neither even, nor odd
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