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Plotting a function graph/ sin5x

Graphing y = sin5x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(5*x)
f(x)=sin⁡(5x)f{\left(x \right)} = \sin{\left(5 x \right)} 
f = sin(5*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⁡(5x)=0\sin{\left(5 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π5x_{2} = \frac{\pi}{5}
Numerical solution
x1=−42.0973415581032x_{1} = -42.0973415581032
x2=45.867252742411x_{2} = 45.867252742411
x3=−15.707963267949x_{3} = -15.707963267949
x4=−57.8053048260522x_{4} = -57.8053048260522
x5=55.9203492338983x_{5} = 55.9203492338983
x6=8.16814089933346x_{6} = 8.16814089933346
x7=26.3893782901543x_{7} = 26.3893782901543
x8=21.9911485751286x_{8} = 21.9911485751286
x9=0x_{9} = 0
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=−64.0884901332318x_{11} = -64.0884901332318
x12=28.2743338823081x_{12} = 28.2743338823081
x13=48.3805268652828x_{13} = 48.3805268652828
x14=−93.6194610769758x_{14} = -93.6194610769758
x15=72.2566310325652x_{15} = 72.2566310325652
x16=−35.8141562509236x_{16} = -35.8141562509236
x17=−47.7522083345649x_{17} = -47.7522083345649
x18=−3.76991118430775x_{18} = -3.76991118430775
x19=−76.026542216873x_{19} = -76.026542216873
x20=87.9645943005142x_{20} = 87.9645943005142
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=50.2654824574367x_{22} = 50.2654824574367
x23=−74.1415866247191x_{23} = -74.1415866247191
x24=16.3362817986669x_{24} = 16.3362817986669
x25=189.752196276824x_{25} = 189.752196276824
x26=−89.8495498926681x_{26} = -89.8495498926681
x27=52.1504380495906x_{27} = 52.1504380495906
x28=18.2212373908208x_{28} = 18.2212373908208
x29=−98.0176907920015x_{29} = -98.0176907920015
x30=−81.6814089933346x_{30} = -81.6814089933346
x31=−77.9114978090269x_{31} = -77.9114978090269
x32=94.2477796076938x_{32} = 94.2477796076938
x33=38.3274303737955x_{33} = 38.3274303737955
x34=−91.734505484822x_{34} = -91.734505484822
x35=−20.1061929829747x_{35} = -20.1061929829747
x36=−32.0442450666159x_{36} = -32.0442450666159
x37=67.8584013175395x_{37} = 67.8584013175395
x38=30.159289474462x_{38} = 30.159289474462
x39=−13.8230076757951x_{39} = -13.8230076757951
x40=−10.0530964914873x_{40} = -10.0530964914873
x41=42.0973415581032x_{41} = 42.0973415581032
x42=−33.9292006587698x_{42} = -33.9292006587698
x43=−79.7964534011807x_{43} = -79.7964534011807
x44=−23.8761041672824x_{44} = -23.8761041672824
x45=65.9734457253857x_{45} = 65.9734457253857
x46=101.159283445591x_{46} = 101.159283445591
x47=98.0176907920015x_{47} = 98.0176907920015
x48=70.3716754404114x_{48} = 70.3716754404114
x49=20.1061929829747x_{49} = 20.1061929829747
x50=−54.0353936417444x_{50} = -54.0353936417444
x51=62.2035345410779x_{51} = 62.2035345410779
x52=32.0442450666159x_{52} = 32.0442450666159
x53=54.0353936417444x_{53} = 54.0353936417444
x54=−87.9645943005142x_{54} = -87.9645943005142
x55=77.9114978090269x_{55} = 77.9114978090269
x56=5.02654824574367x_{56} = 5.02654824574367
x57=−27.6460153515902x_{57} = -27.6460153515902
x58=99.9026463841554x_{58} = 99.9026463841554
x59=60.318578948924x_{59} = 60.318578948924
x60=11.3097335529233x_{60} = 11.3097335529233
x61=64.0884901332318x_{61} = 64.0884901332318
x62=11.9380520836412x_{62} = 11.9380520836412
x63=−11.9380520836412x_{63} = -11.9380520836412
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=92.3628240155399x_{65} = 92.3628240155399
x66=43.9822971502571x_{66} = 43.9822971502571
x67=84.1946831162065x_{67} = 84.1946831162065
x68=−49.6371639267187x_{68} = -49.6371639267187
x69=−37.6991118430775x_{69} = -37.6991118430775
x70=−59.6902604182061x_{70} = -59.6902604182061
x71=33.9292006587698x_{71} = 33.9292006587698
x72=86.0796387083603x_{72} = 86.0796387083603
x73=40.2123859659494x_{73} = 40.2123859659494
x74=−96.1327351998477x_{74} = -96.1327351998477
x75=82.3097275240526x_{75} = 82.3097275240526
x76=−5.65486677646163x_{76} = -5.65486677646163
x77=89.2212313619501x_{77} = 89.2212313619501
x78=76.026542216873x_{78} = 76.026542216873
x79=−71.6283125018473x_{79} = -71.6283125018473
x80=−55.9203492338983x_{80} = -55.9203492338983
x81=6.28318530717959x_{81} = 6.28318530717959
x82=−45.867252742411x_{82} = -45.867252742411
x83=−69.7433569096934x_{83} = -69.7433569096934
x84=−1.88495559215388x_{84} = -1.88495559215388
x85=74.1415866247191x_{85} = 74.1415866247191
x86=10.0530964914873x_{86} = 10.0530964914873
x87=1.88495559215388x_{87} = 1.88495559215388
x88=23.8761041672824x_{88} = 23.8761041672824
x89=−25.7610597594363x_{89} = -25.7610597594363
x90=52.7787565803085x_{90} = 52.7787565803085
x91=5.65486677646163x_{91} = 5.65486677646163
x92=−99.9026463841554x_{92} = -99.9026463841554
x93=−86.0796387083603x_{93} = -86.0796387083603
x94=−67.8584013175395x_{94} = -67.8584013175395
x95=−52.7787565803085x_{95} = -52.7787565803085
x96=96.1327351998477x_{96} = 96.1327351998477
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(5*x).
sin⁡(0⋅5)\sin{\left(0 \cdot 5 \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
5cos⁡(5x)=05 \cos{\left(5 x \right)} = 0
Solve this equation
The roots of this equation
x1=π10x_{1} = \frac{\pi}{10}
x2=3π10x_{2} = \frac{3 \pi}{10}
The values of the extrema at the points:
 pi    
(--, 1)
 10    

 3*pi     
(----, -1)
  10


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=3π10x_{1} = \frac{3 \pi}{10}
Maxima of the function at points:
x1=π10x_{1} = \frac{\pi}{10}
Decreasing at intervals
(−∞,π10]∪[3π10,∞)\left(-\infty, \frac{\pi}{10}\right] \cup \left[\frac{3 \pi}{10}, \infty\right)
Increasing at intervals
[π10,3π10]\left[\frac{\pi}{10}, \frac{3 \pi}{10}\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
−25sin⁡(5x)=0- 25 \sin{\left(5 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π5x_{2} = \frac{\pi}{5}

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]∪[π5,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{5}, \infty\right)
Convex at the intervals
[0,π5]\left[0, \frac{\pi}{5}\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⁡(5x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(5 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⁡(5x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(5 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(5*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(5 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⁡(5x)=−sin⁡(5x)\sin{\left(5 x \right)} = - \sin{\left(5 x \right)}
- No
sin⁡(5x)=sin⁡(5x)\sin{\left(5 x \right)} = \sin{\left(5 x \right)}
- Yes
so, the function
is
odd
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