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Plotting a function graph/ 4sinx

Graphing y = 4sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 4*sin(x)
f(x)=4sin⁡(x)f{\left(x \right)} = 4 \sin{\left(x \right)} 
f = 4*sin(x)
The graph of the function
02468-8-6-4-2-1010-1010
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:
4sin⁡(x)=04 \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=−18.8495559215388x_{1} = -18.8495559215388
x2=−53.4070751110265x_{2} = -53.4070751110265
x3=−37.6991118430775x_{3} = -37.6991118430775
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−15.707963267949x_{5} = -15.707963267949
x6=−113.097335529233x_{6} = -113.097335529233
x7=−56.5486677646163x_{7} = -56.5486677646163
x8=12.5663706143592x_{8} = 12.5663706143592
x9=3.14159265358979x_{9} = 3.14159265358979
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=84.8230016469244x_{11} = 84.8230016469244
x12=−81.6814089933346x_{12} = -81.6814089933346
x13=94.2477796076938x_{13} = 94.2477796076938
x14=21.9911485751286x_{14} = 21.9911485751286
x15=0x_{15} = 0
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=81.6814089933346x_{17} = 81.6814089933346
x18=40.8407044966673x_{18} = 40.8407044966673
x19=−75.398223686155x_{19} = -75.398223686155
x20=−78.5398163397448x_{20} = -78.5398163397448
x21=62.8318530717959x_{21} = 62.8318530717959
x22=100.530964914873x_{22} = 100.530964914873
x23=−21.9911485751286x_{23} = -21.9911485751286
x24=47.1238898038469x_{24} = 47.1238898038469
x25=91.106186954104x_{25} = 91.106186954104
x26=−232.477856365645x_{26} = -232.477856365645
x27=75.398223686155x_{27} = 75.398223686155
x28=28.2743338823081x_{28} = 28.2743338823081
x29=34.5575191894877x_{29} = 34.5575191894877
x30=6.28318530717959x_{30} = 6.28318530717959
x31=78.5398163397448x_{31} = 78.5398163397448
x32=72.2566310325652x_{32} = 72.2566310325652
x33=−6.28318530717959x_{33} = -6.28318530717959
x34=15.707963267949x_{34} = 15.707963267949
x35=31.4159265358979x_{35} = 31.4159265358979
x36=−47.1238898038469x_{36} = -47.1238898038469
x37=25.1327412287183x_{37} = 25.1327412287183
x38=18.8495559215388x_{38} = 18.8495559215388
x39=−94.2477796076938x_{39} = -94.2477796076938
x40=−3.14159265358979x_{40} = -3.14159265358979
x41=−40.8407044966673x_{41} = -40.8407044966673
x42=56.5486677646163x_{42} = 56.5486677646163
x43=−25.1327412287183x_{43} = -25.1327412287183
x44=53.4070751110265x_{44} = 53.4070751110265
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=−2642.07942166902x_{46} = -2642.07942166902
x47=−9.42477796076938x_{47} = -9.42477796076938
x48=87.9645943005142x_{48} = 87.9645943005142
x49=−50.2654824574367x_{49} = -50.2654824574367
x50=−100.530964914873x_{50} = -100.530964914873
x51=−43.9822971502571x_{51} = -43.9822971502571
x52=50.2654824574367x_{52} = 50.2654824574367
x53=−97.3893722612836x_{53} = -97.3893722612836
x54=69.1150383789755x_{54} = 69.1150383789755
x55=59.6902604182061x_{55} = 59.6902604182061
x56=97.3893722612836x_{56} = 97.3893722612836
x57=−62.8318530717959x_{57} = -62.8318530717959
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=−91.106186954104x_{59} = -91.106186954104
x60=−12.5663706143592x_{60} = -12.5663706143592
x61=−69.1150383789755x_{61} = -69.1150383789755
x62=37.6991118430775x_{62} = 37.6991118430775
x63=9.42477796076938x_{63} = 9.42477796076938
x64=65.9734457253857x_{64} = 65.9734457253857
x65=−65.9734457253857x_{65} = -65.9734457253857
x66=−267.035375555132x_{66} = -267.035375555132
x67=−84.8230016469244x_{67} = -84.8230016469244
x68=−34.5575191894877x_{68} = -34.5575191894877
x69=43.9822971502571x_{69} = 43.9822971502571
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4*sin(x).
4sin⁡(0)4 \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
4cos⁡(x)=04 \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    
(--, 4)
 2     

 3*pi     
(----, -4)
  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
−4sin⁡(x)=0- 4 \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→−∞(4sin⁡(x))=⟨−4,4⟩\lim_{x \to -\infty}\left(4 \sin{\left(x \right)}\right) = \left\langle -4, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−4,4⟩y = \left\langle -4, 4\right\rangle
lim⁡x→∞(4sin⁡(x))=⟨−4,4⟩\lim_{x \to \infty}\left(4 \sin{\left(x \right)}\right) = \left\langle -4, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−4,4⟩y = \left\langle -4, 4\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(4sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{4 \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(4sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{4 \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:
4sin⁡(x)=−4sin⁡(x)4 \sin{\left(x \right)} = - 4 \sin{\left(x \right)}
- No
4sin⁡(x)=4sin⁡(x)4 \sin{\left(x \right)} = 4 \sin{\left(x \right)}
- Yes
so, the function
is
odd
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