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Plotting a function graph/ sinx

Graphing y = sinx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(x)
f(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} 
f = sin(x)
The graph of the function
0.000.250.500.751.001.251.501.752.002.252.502.753.0002
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)=0\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=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−2642.07942166902x_{18} = -2642.07942166902
x19=−15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=−267.035375555132x_{21} = -267.035375555132
x22=6.28318530717959x_{22} = 6.28318530717959
x23=9.42477796076938x_{23} = 9.42477796076938
x24=−53.4070751110265x_{24} = -53.4070751110265
x25=−47.1238898038469x_{25} = -47.1238898038469
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=69.1150383789755x_{27} = 69.1150383789755
x28=21.9911485751286x_{28} = 21.9911485751286
x29=87.9645943005142x_{29} = 87.9645943005142
x30=18.8495559215388x_{30} = 18.8495559215388
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=−72.2566310325652x_{32} = -72.2566310325652
x33=25.1327412287183x_{33} = 25.1327412287183
x34=37.6991118430775x_{34} = 37.6991118430775
x35=−25.1327412287183x_{35} = -25.1327412287183
x36=0x_{36} = 0
x37=50.2654824574367x_{37} = 50.2654824574367
x38=−6.28318530717959x_{38} = -6.28318530717959
x39=−65.9734457253857x_{39} = -65.9734457253857
x40=−21.9911485751286x_{40} = -21.9911485751286
x41=−62.8318530717959x_{41} = -62.8318530717959
x42=75.398223686155x_{42} = 75.398223686155
x43=84.8230016469244x_{43} = 84.8230016469244
x44=53.4070751110265x_{44} = 53.4070751110265
x45=34.5575191894877x_{45} = 34.5575191894877
x46=−28.2743338823081x_{46} = -28.2743338823081
x47=15.707963267949x_{47} = 15.707963267949
x48=−91.106186954104x_{48} = -91.106186954104
x49=47.1238898038469x_{49} = 47.1238898038469
x50=97.3893722612836x_{50} = 97.3893722612836
x51=−69.1150383789755x_{51} = -69.1150383789755
x52=−232.477856365645x_{52} = -232.477856365645
x53=94.2477796076938x_{53} = 94.2477796076938
x54=−18.8495559215388x_{54} = -18.8495559215388
x55=−50.2654824574367x_{55} = -50.2654824574367
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=−81.6814089933346x_{57} = -81.6814089933346
x58=62.8318530717959x_{58} = 62.8318530717959
x59=78.5398163397448x_{59} = 78.5398163397448
x60=31.4159265358979x_{60} = 31.4159265358979
x61=−78.5398163397448x_{61} = -78.5398163397448
x62=−40.8407044966673x_{62} = -40.8407044966673
x63=−97.3893722612836x_{63} = -97.3893722612836
x64=−113.097335529233x_{64} = -113.097335529233
x65=−75.398223686155x_{65} = -75.398223686155
x66=91.106186954104x_{66} = 91.106186954104
x67=−12.5663706143592x_{67} = -12.5663706143592
x68=−94.2477796076938x_{68} = -94.2477796076938
x69=−34.5575191894877x_{69} = -34.5575191894877
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⁡(0)\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)=0\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    
(--, 1)
 2     

 3*pi     
(----, -1)
  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
−sin⁡(x)=0- \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→−∞sin⁡(x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\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:
sin⁡(x)=−sin⁡(x)\sin{\left(x \right)} = - \sin{\left(x \right)}
- No
sin⁡(x)=sin⁡(x)\sin{\left(x \right)} = \sin{\left(x \right)}
- Yes
so, the function
is
odd
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