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Plotting a function graph/ 8*sin(x)

Graphing y = 8*sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 8*sin(x)
f(x)=8sin⁡(x)f{\left(x \right)} = 8 \sin{\left(x \right)} 
f = 8*sin(x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
8sin⁡(x)=08 \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=62.8318530717959x_{1} = 62.8318530717959
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=47.1238898038469x_{3} = 47.1238898038469
x4=84.8230016469244x_{4} = 84.8230016469244
x5=−53.4070751110265x_{5} = -53.4070751110265
x6=91.106186954104x_{6} = 91.106186954104
x7=−84.8230016469244x_{7} = -84.8230016469244
x8=−2642.07942166902x_{8} = -2642.07942166902
x9=25.1327412287183x_{9} = 25.1327412287183
x10=−3.14159265358979x_{10} = -3.14159265358979
x11=−6.28318530717959x_{11} = -6.28318530717959
x12=−40.8407044966673x_{12} = -40.8407044966673
x13=−18.8495559215388x_{13} = -18.8495559215388
x14=78.5398163397448x_{14} = 78.5398163397448
x15=−75.398223686155x_{15} = -75.398223686155
x16=−9.42477796076938x_{16} = -9.42477796076938
x17=72.2566310325652x_{17} = 72.2566310325652
x18=−43.9822971502571x_{18} = -43.9822971502571
x19=31.4159265358979x_{19} = 31.4159265358979
x20=9.42477796076938x_{20} = 9.42477796076938
x21=−267.035375555132x_{21} = -267.035375555132
x22=40.8407044966673x_{22} = 40.8407044966673
x23=−69.1150383789755x_{23} = -69.1150383789755
x24=12.5663706143592x_{24} = 12.5663706143592
x25=87.9645943005142x_{25} = 87.9645943005142
x26=59.6902604182061x_{26} = 59.6902604182061
x27=−37.6991118430775x_{27} = -37.6991118430775
x28=−100.530964914873x_{28} = -100.530964914873
x29=−91.106186954104x_{29} = -91.106186954104
x30=97.3893722612836x_{30} = 97.3893722612836
x31=0x_{31} = 0
x32=−12.5663706143592x_{32} = -12.5663706143592
x33=−78.5398163397448x_{33} = -78.5398163397448
x34=−232.477856365645x_{34} = -232.477856365645
x35=18.8495559215388x_{35} = 18.8495559215388
x36=−94.2477796076938x_{36} = -94.2477796076938
x37=34.5575191894877x_{37} = 34.5575191894877
x38=−113.097335529233x_{38} = -113.097335529233
x39=43.9822971502571x_{39} = 43.9822971502571
x40=−31.4159265358979x_{40} = -31.4159265358979
x41=−81.6814089933346x_{41} = -81.6814089933346
x42=−65.9734457253857x_{42} = -65.9734457253857
x43=75.398223686155x_{43} = 75.398223686155
x44=56.5486677646163x_{44} = 56.5486677646163
x45=3.14159265358979x_{45} = 3.14159265358979
x46=15.707963267949x_{46} = 15.707963267949
x47=−56.5486677646163x_{47} = -56.5486677646163
x48=−21.9911485751286x_{48} = -21.9911485751286
x49=50.2654824574367x_{49} = 50.2654824574367
x50=−15.707963267949x_{50} = -15.707963267949
x51=28.2743338823081x_{51} = 28.2743338823081
x52=94.2477796076938x_{52} = 94.2477796076938
x53=−59.6902604182061x_{53} = -59.6902604182061
x54=−62.8318530717959x_{54} = -62.8318530717959
x55=69.1150383789755x_{55} = 69.1150383789755
x56=−34.5575191894877x_{56} = -34.5575191894877
x57=−97.3893722612836x_{57} = -97.3893722612836
x58=21.9911485751286x_{58} = 21.9911485751286
x59=65.9734457253857x_{59} = 65.9734457253857
x60=37.6991118430775x_{60} = 37.6991118430775
x61=−87.9645943005142x_{61} = -87.9645943005142
x62=−72.2566310325652x_{62} = -72.2566310325652
x63=−25.1327412287183x_{63} = -25.1327412287183
x64=−28.2743338823081x_{64} = -28.2743338823081
x65=81.6814089933346x_{65} = 81.6814089933346
x66=6.28318530717959x_{66} = 6.28318530717959
x67=100.530964914873x_{67} = 100.530964914873
x68=53.4070751110265x_{68} = 53.4070751110265
x69=−47.1238898038469x_{69} = -47.1238898038469
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 8*sin(x).
8sin⁡(0)8 \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
8cos⁡(x)=08 \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    
(--, 8)
 2     

 3*pi     
(----, -8)
  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
−8sin⁡(x)=0- 8 \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→−∞(8sin⁡(x))=⟨−8,8⟩\lim_{x \to -\infty}\left(8 \sin{\left(x \right)}\right) = \left\langle -8, 8\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−8,8⟩y = \left\langle -8, 8\right\rangle
lim⁡x→∞(8sin⁡(x))=⟨−8,8⟩\lim_{x \to \infty}\left(8 \sin{\left(x \right)}\right) = \left\langle -8, 8\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−8,8⟩y = \left\langle -8, 8\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 8*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(8sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{8 \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(8sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{8 \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:
8sin⁡(x)=−8sin⁡(x)8 \sin{\left(x \right)} = - 8 \sin{\left(x \right)}
- No
8sin⁡(x)=8sin⁡(x)8 \sin{\left(x \right)} = 8 \sin{\left(x \right)}
- Yes
so, the function
is
odd
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