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Plotting a function graph/ sin(10*x)

Graphing y = sin(10*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(10*x)
f(x)=sin⁡(10x)f{\left(x \right)} = \sin{\left(10 x \right)} 
f = sin(10*x)
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⁡(10x)=0\sin{\left(10 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π10x_{2} = \frac{\pi}{10}
Numerical solution
x1=−61.8893752757189x_{1} = -61.8893752757189
x2=14.1371669411541x_{2} = 14.1371669411541
x3=−3.76991118430775x_{3} = -3.76991118430775
x4=−63.7743308678728x_{4} = -63.7743308678728
x5=60.0044196835651x_{5} = 60.0044196835651
x6=40.2123859659494x_{6} = 40.2123859659494
x7=72.2566310325652x_{7} = 72.2566310325652
x8=−43.9822971502571x_{8} = -43.9822971502571
x9=−13.8230076757951x_{9} = -13.8230076757951
x10=24.1902634326414x_{10} = 24.1902634326414
x11=−89.8495498926681x_{11} = -89.8495498926681
x12=16.0221225333079x_{12} = 16.0221225333079
x13=−27.9601746169492x_{13} = -27.9601746169492
x14=−31.7300858012569x_{14} = -31.7300858012569
x15=−49.9513231920777x_{15} = -49.9513231920777
x16=68.1725605828985x_{16} = 68.1725605828985
x17=−51.8362787842316x_{17} = -51.8362787842316
x18=30.159289474462x_{18} = 30.159289474462
x19=−1.88495559215388x_{19} = -1.88495559215388
x20=−21.9911485751286x_{20} = -21.9911485751286
x21=50.2654824574367x_{21} = 50.2654824574367
x22=−9.73893722612836x_{22} = -9.73893722612836
x23=78.2256570743859x_{23} = 78.2256570743859
x24=−33.9292006587698x_{24} = -33.9292006587698
x25=−35.8141562509236x_{25} = -35.8141562509236
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=21.9911485751286x_{27} = 21.9911485751286
x28=−69.7433569096934x_{28} = -69.7433569096934
x29=−25.7610597594363x_{29} = -25.7610597594363
x30=90.1637091580271x_{30} = 90.1637091580271
x31=12.2522113490002x_{31} = 12.2522113490002
x32=−47.7522083345649x_{32} = -47.7522083345649
x33=70.0575161750524x_{33} = 70.0575161750524
x34=20.1061929829747x_{34} = 20.1061929829747
x35=2.19911485751286x_{35} = 2.19911485751286
x36=−5.96902604182061x_{36} = -5.96902604182061
x37=−57.8053048260522x_{37} = -57.8053048260522
x38=48.0663675999238x_{38} = 48.0663675999238
x39=26.0752190247953x_{39} = 26.0752190247953
x40=−98.0176907920015x_{40} = -98.0176907920015
x41=8.16814089933346x_{41} = 8.16814089933346
x42=−29.845130209103x_{42} = -29.845130209103
x43=32.0442450666159x_{43} = 32.0442450666159
x44=−60.0044196835651x_{44} = -60.0044196835651
x45=−99.9026463841554x_{45} = -99.9026463841554
x46=10.0530964914873x_{46} = 10.0530964914873
x47=−81.9955682586936x_{47} = -81.9955682586936
x48=4.08407044966673x_{48} = 4.08407044966673
x49=56.2345084992573x_{49} = 56.2345084992573
x50=−11.9380520836412x_{50} = -11.9380520836412
x51=−65.9734457253857x_{51} = -65.9734457253857
x52=96.1327351998477x_{52} = 96.1327351998477
x53=94.2477796076938x_{53} = 94.2477796076938
x54=−71.9424717672063x_{54} = -71.9424717672063
x55=37.6991118430775x_{55} = 37.6991118430775
x56=−19.7920337176157x_{56} = -19.7920337176157
x57=−53.4070751110265x_{57} = -53.4070751110265
x58=84.1946831162065x_{58} = 84.1946831162065
x59=−39.8982267005904x_{59} = -39.8982267005904
x60=−77.9114978090269x_{60} = -77.9114978090269
x61=36.1283155162826x_{61} = 36.1283155162826
x62=18.2212373908208x_{62} = 18.2212373908208
x63=−45.867252742411x_{63} = -45.867252742411
x64=−93.9336203423348x_{64} = -93.9336203423348
x65=76.026542216873x_{65} = 76.026542216873
x66=80.1106126665397x_{66} = 80.1106126665397
x67=−79.7964534011807x_{67} = -79.7964534011807
x68=0x_{68} = 0
x69=−23.8761041672824x_{69} = -23.8761041672824
x70=54.0353936417444x_{70} = 54.0353936417444
x71=58.1194640914112x_{71} = 58.1194640914112
x72=−38.0132711084365x_{72} = -38.0132711084365
x73=52.1504380495906x_{73} = 52.1504380495906
x74=100.216805649514x_{74} = 100.216805649514
x75=−16.0221225333079x_{75} = -16.0221225333079
x76=92.0486647501809x_{76} = 92.0486647501809
x77=64.0884901332318x_{77} = 64.0884901332318
x78=98.0176907920015x_{78} = 98.0176907920015
x79=−95.8185759344887x_{79} = -95.8185759344887
x80=34.2433599241287x_{80} = 34.2433599241287
x81=−41.7831822927443x_{81} = -41.7831822927443
x82=46.18141200777x_{82} = 46.18141200777
x83=−17.9070781254618x_{83} = -17.9070781254618
x84=86.0796387083603x_{84} = 86.0796387083603
x85=38.0132711084365x_{85} = 38.0132711084365
x86=−55.9203492338983x_{86} = -55.9203492338983
x87=62.2035345410779x_{87} = 62.2035345410779
x88=−83.8805238508475x_{88} = -83.8805238508475
x89=−91.734505484822x_{89} = -91.734505484822
x90=74.1415866247191x_{90} = 74.1415866247191
x91=43.9822971502571x_{91} = 43.9822971502571
x92=42.0973415581032x_{92} = 42.0973415581032
x93=−67.8584013175395x_{93} = -67.8584013175395
x94=−73.8274273593601x_{94} = -73.8274273593601
x95=81.9955682586936x_{95} = 81.9955682586936
x96=−136.659280431156x_{96} = -136.659280431156
x97=−85.7654794430014x_{97} = -85.7654794430014
x98=−7.85398163397448x_{98} = -7.85398163397448
x99=88.5929128312322x_{99} = 88.5929128312322
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(10*x).
sin⁡(0⋅10)\sin{\left(0 \cdot 10 \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
10cos⁡(10x)=010 \cos{\left(10 x \right)} = 0
Solve this equation
The roots of this equation
x1=π20x_{1} = \frac{\pi}{20}
x2=3π20x_{2} = \frac{3 \pi}{20}
The values of the extrema at the points:
 pi    
(--, 1)
 20    

 3*pi     
(----, -1)
  20


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=3π20x_{1} = \frac{3 \pi}{20}
Maxima of the function at points:
x1=π20x_{1} = \frac{\pi}{20}
Decreasing at intervals
(−∞,π20]∪[3π20,∞)\left(-\infty, \frac{\pi}{20}\right] \cup \left[\frac{3 \pi}{20}, \infty\right)
Increasing at intervals
[π20,3π20]\left[\frac{\pi}{20}, \frac{3 \pi}{20}\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
−100sin⁡(10x)=0- 100 \sin{\left(10 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π10x_{2} = \frac{\pi}{10}

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]∪[π10,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{10}, \infty\right)
Convex at the intervals
[0,π10]\left[0, \frac{\pi}{10}\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⁡(10x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(10 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⁡(10x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(10 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(10*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(10x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(10 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(10x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(10 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⁡(10x)=−sin⁡(10x)\sin{\left(10 x \right)} = - \sin{\left(10 x \right)}
- No
sin⁡(10x)=sin⁡(10x)\sin{\left(10 x \right)} = \sin{\left(10 x \right)}
- Yes
so, the function
is
odd
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