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Plotting a function graph/ sin(5*x)

Graphing y = sin(5*x)

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