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Plotting a function graph/ sin(6*x)

Graphing y = sin(6*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(6*x)
f(x)=sin⁡(6x)f{\left(x \right)} = \sin{\left(6 x \right)} 
f = sin(6*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⁡(6x)=0\sin{\left(6 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π6x_{2} = \frac{\pi}{6}
Numerical solution
x1=82.2050077689329x_{1} = 82.2050077689329
x2=62.3082542961976x_{2} = 62.3082542961976
x3=97.9129710368819x_{3} = 97.9129710368819
x4=−41.8879020478639x_{4} = -41.8879020478639
x5=14.1371669411541x_{5} = 14.1371669411541
x6=−68.0678408277789x_{6} = -68.0678408277789
x7=72.2566310325652x_{7} = 72.2566310325652
x8=−43.9822971502571x_{8} = -43.9822971502571
x9=−90.0589894029074x_{9} = -90.0589894029074
x10=9.94837673636768x_{10} = 9.94837673636768
x11=12.0427718387609x_{11} = 12.0427718387609
x12=−37.6991118430775x_{12} = -37.6991118430775
x13=34.5575191894877x_{13} = 34.5575191894877
x14=−49.7418836818384x_{14} = -49.7418836818384
x15=−75.9218224617533x_{15} = -75.9218224617533
x16=−51.8362787842316x_{16} = -51.8362787842316
x17=−21.9911485751286x_{17} = -21.9911485751286
x18=50.2654824574367x_{18} = 50.2654824574367
x19=−15.707963267949x_{19} = -15.707963267949
x20=−46.0766922526503x_{20} = -46.0766922526503
x21=−2.0943951023932x_{21} = -2.0943951023932
x22=67.5442420521806x_{22} = 67.5442420521806
x23=21.9911485751286x_{23} = 21.9911485751286
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=−90.5825881785057x_{25} = -90.5825881785057
x26=70.162235930172x_{26} = 70.162235930172
x27=4.18879020478639x_{27} = 4.18879020478639
x28=38.2227106186758x_{28} = 38.2227106186758
x29=34.0339204138894x_{29} = 34.0339204138894
x30=−31.9395253114962x_{30} = -31.9395253114962
x31=60.2138591938044x_{31} = 60.2138591938044
x32=−59.1666616426078x_{32} = -59.1666616426078
x33=−85.870199198121x_{33} = -85.870199198121
x34=−29.845130209103x_{34} = -29.845130209103
x35=−78.0162175641465x_{35} = -78.0162175641465
x36=−384.84510006475x_{36} = -384.84510006475
x37=48.1710873550435x_{37} = 48.1710873550435
x38=−17.8023583703422x_{38} = -17.8023583703422
x39=10.471975511966x_{39} = 10.471975511966
x40=−93.7241808320955x_{40} = -93.7241808320955
x41=−65.9734457253857x_{41} = -65.9734457253857
x42=46.0766922526503x_{42} = 46.0766922526503
x43=−83.7758040957278x_{43} = -83.7758040957278
x44=28.2743338823081x_{44} = 28.2743338823081
x45=94.2477796076938x_{45} = 94.2477796076938
x46=75.9218224617533x_{46} = 75.9218224617533
x47=96.342174710087x_{47} = 96.342174710087
x48=−82.2050077689329x_{48} = -82.2050077689329
x49=6.28318530717959x_{49} = 6.28318530717959
x50=−25.6563400043166x_{50} = -25.6563400043166
x51=93.2005820564972x_{51} = 93.2005820564972
x52=36.1283155162826x_{52} = 36.1283155162826
x53=−19.8967534727354x_{53} = -19.8967534727354
x54=−12.0427718387609x_{54} = -12.0427718387609
x55=80.1106126665397x_{55} = 80.1106126665397
x56=−61.7846555205993x_{56} = -61.7846555205993
x57=56.025068989018x_{57} = 56.025068989018
x58=−24.0855436775217x_{58} = -24.0855436775217
x59=−56.025068989018x_{59} = -56.025068989018
x60=78.0162175641465x_{60} = 78.0162175641465
x61=0x_{61} = 0
x62=18.3259571459405x_{62} = 18.3259571459405
x63=58.1194640914112x_{63} = 58.1194640914112
x64=−39.7935069454707x_{64} = -39.7935069454707
x65=−9.94837673636768x_{65} = -9.94837673636768
x66=2.0943951023932x_{66} = 2.0943951023932
x67=−34.0339204138894x_{67} = -34.0339204138894
x68=−100.007366139275x_{68} = -100.007366139275
x69=−95.8185759344887x_{69} = -95.8185759344887
x70=24.0855436775217x_{70} = 24.0855436775217
x71=99.4837673636768x_{71} = 99.4837673636768
x72=40.317105721069x_{72} = 40.317105721069
x73=−97.9129710368819x_{73} = -97.9129710368819
x74=100.007366139275x_{74} = 100.007366139275
x75=−53.9306738866248x_{75} = -53.9306738866248
x76=31.9395253114962x_{76} = 31.9395253114962
x77=−3.66519142918809x_{77} = -3.66519142918809
x78=92.1533845053006x_{78} = 92.1533845053006
x79=87.9645943005142x_{79} = 87.9645943005142
x80=84.2994028713261x_{80} = 84.2994028713261
x81=53.9306738866248x_{81} = 53.9306738866248
x82=43.9822971502571x_{82} = 43.9822971502571
x83=−81.6814089933346x_{83} = -81.6814089933346
x84=−63.8790506229925x_{84} = -63.8790506229925
x85=−73.8274273593601x_{85} = -73.8274273593601
x86=68.0678408277789x_{86} = 68.0678408277789
x87=26.1799387799149x_{87} = 26.1799387799149
x88=16.2315620435473x_{88} = 16.2315620435473
x89=−71.733032256967x_{89} = -71.733032256967
x90=−59.6902604182061x_{90} = -59.6902604182061
x91=65.9734457253857x_{91} = 65.9734457253857
x92=90.0589894029074x_{92} = 90.0589894029074
x93=−7.85398163397448x_{93} = -7.85398163397448
x94=−5.75958653158129x_{94} = -5.75958653158129
x95=−27.7507351067098x_{95} = -27.7507351067098
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(6*x).
sin⁡(0⋅6)\sin{\left(0 \cdot 6 \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
6cos⁡(6x)=06 \cos{\left(6 x \right)} = 0
Solve this equation
The roots of this equation
x1=π12x_{1} = \frac{\pi}{12}
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
 pi    
(--, 1)
 12    

 pi     
(--, -1)
 4


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=π4x_{1} = \frac{\pi}{4}
Maxima of the function at points:
x1=π12x_{1} = \frac{\pi}{12}
Decreasing at intervals
(−∞,π12]∪[π4,∞)\left(-\infty, \frac{\pi}{12}\right] \cup \left[\frac{\pi}{4}, \infty\right)
Increasing at intervals
[π12,π4]\left[\frac{\pi}{12}, \frac{\pi}{4}\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
−36sin⁡(6x)=0- 36 \sin{\left(6 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π6x_{2} = \frac{\pi}{6}

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]∪[π6,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{6}, \infty\right)
Convex at the intervals
[0,π6]\left[0, \frac{\pi}{6}\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⁡(6x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(6 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⁡(6x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(6 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(6*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(6x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(6 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(6x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(6 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⁡(6x)=−sin⁡(6x)\sin{\left(6 x \right)} = - \sin{\left(6 x \right)}
- No
sin⁡(6x)=sin⁡(6x)\sin{\left(6 x \right)} = \sin{\left(6 x \right)}
- Yes
so, the function
is
odd
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