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2sinx
Plotting a function graph/ 2sinx

Graphing y = 2sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(x)
f(x)=2sin⁡(x)f{\left(x \right)} = 2 \sin{\left(x \right)} 
f = 2*sin(x)
The graph of the function
0-50-40-30-20-101020304050605-5
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:
2sin⁡(x)=02 \sin{\left(x \right)} = 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=69.1150383789755x_{1} = 69.1150383789755
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=−25.1327412287183x_{3} = -25.1327412287183
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=97.3893722612836x_{6} = 97.3893722612836
x7=−81.6814089933346x_{7} = -81.6814089933346
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=62.8318530717959x_{9} = 62.8318530717959
x10=−69.1150383789755x_{10} = -69.1150383789755
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−56.5486677646163x_{12} = -56.5486677646163
x13=37.6991118430775x_{13} = 37.6991118430775
x14=25.1327412287183x_{14} = 25.1327412287183
x15=−78.5398163397448x_{15} = -78.5398163397448
x16=−62.8318530717959x_{16} = -62.8318530717959
x17=−232.477856365645x_{17} = -232.477856365645
x18=−113.097335529233x_{18} = -113.097335529233
x19=−65.9734457253857x_{19} = -65.9734457253857
x20=−87.9645943005142x_{20} = -87.9645943005142
x21=−2642.07942166902x_{21} = -2642.07942166902
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=21.9911485751286x_{23} = 21.9911485751286
x24=−47.1238898038469x_{24} = -47.1238898038469
x25=−100.530964914873x_{25} = -100.530964914873
x26=6.28318530717959x_{26} = 6.28318530717959
x27=−75.398223686155x_{27} = -75.398223686155
x28=−72.2566310325652x_{28} = -72.2566310325652
x29=9.42477796076938x_{29} = 9.42477796076938
x30=56.5486677646163x_{30} = 56.5486677646163
x31=65.9734457253857x_{31} = 65.9734457253857
x32=100.530964914873x_{32} = 100.530964914873
x33=−28.2743338823081x_{33} = -28.2743338823081
x34=43.9822971502571x_{34} = 43.9822971502571
x35=−6.28318530717959x_{35} = -6.28318530717959
x36=31.4159265358979x_{36} = 31.4159265358979
x37=−267.035375555132x_{37} = -267.035375555132
x38=−3.14159265358979x_{38} = -3.14159265358979
x39=28.2743338823081x_{39} = 28.2743338823081
x40=−37.6991118430775x_{40} = -37.6991118430775
x41=−84.8230016469244x_{41} = -84.8230016469244
x42=84.8230016469244x_{42} = 84.8230016469244
x43=−34.5575191894877x_{43} = -34.5575191894877
x44=−91.106186954104x_{44} = -91.106186954104
x45=15.707963267949x_{45} = 15.707963267949
x46=94.2477796076938x_{46} = 94.2477796076938
x47=75.398223686155x_{47} = 75.398223686155
x48=34.5575191894877x_{48} = 34.5575191894877
x49=91.106186954104x_{49} = 91.106186954104
x50=−18.8495559215388x_{50} = -18.8495559215388
x51=−12.5663706143592x_{51} = -12.5663706143592
x52=−15.707963267949x_{52} = -15.707963267949
x53=59.6902604182061x_{53} = 59.6902604182061
x54=−94.2477796076938x_{54} = -94.2477796076938
x55=47.1238898038469x_{55} = 47.1238898038469
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=−9.42477796076938x_{57} = -9.42477796076938
x58=3.14159265358979x_{58} = 3.14159265358979
x59=40.8407044966673x_{59} = 40.8407044966673
x60=−40.8407044966673x_{60} = -40.8407044966673
x61=72.2566310325652x_{61} = 72.2566310325652
x62=0x_{62} = 0
x63=53.4070751110265x_{63} = 53.4070751110265
x64=50.2654824574367x_{64} = 50.2654824574367
x65=−97.3893722612836x_{65} = -97.3893722612836
x66=12.5663706143592x_{66} = 12.5663706143592
x67=78.5398163397448x_{67} = 78.5398163397448
x68=−31.4159265358979x_{68} = -31.4159265358979
x69=18.8495559215388x_{69} = 18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(x).
2sin⁡(0)2 \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
2cos⁡(x)=02 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi    
(--, 2)
 2     

 3*pi     
(----, -2)
  2


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=3π2x_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\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⁡(x)=0- 2 \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi

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]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(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→∞(2sin⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(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 2*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(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:
2sin⁡(x)=−2sin⁡(x)2 \sin{\left(x \right)} = - 2 \sin{\left(x \right)}
- No
2sin⁡(x)=2sin⁡(x)2 \sin{\left(x \right)} = 2 \sin{\left(x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = 2sinx
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