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sin(3x)
Plotting a function graph/ sin(3x)

Graphing y = sin(3x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(3*x)
f(x)=sin⁡(3x)f{\left(x \right)} = \sin{\left(3 x \right)} 
f = sin(3*x)
The graph of the function
010203040506070-102-2
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⁡(3x)=0\sin{\left(3 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π3x_{2} = \frac{\pi}{3}
Numerical solution
x1=−4.18879020478639x_{1} = -4.18879020478639
x2=41.8879020478639x_{2} = 41.8879020478639
x3=72.2566310325652x_{3} = 72.2566310325652
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=17.8023583703422x_{5} = 17.8023583703422
x6=74.3510261349584x_{6} = 74.3510261349584
x7=−39.7935069454707x_{7} = -39.7935069454707
x8=−29.3215314335047x_{8} = -29.3215314335047
x9=−57.5958653158129x_{9} = -57.5958653158129
x10=37.6991118430775x_{10} = 37.6991118430775
x11=−63.8790506229925x_{11} = -63.8790506229925
x12=−13.6135681655558x_{12} = -13.6135681655558
x13=54.4542726622231x_{13} = 54.4542726622231
x14=−55.5014702134197x_{14} = -55.5014702134197
x15=80.634211442138x_{15} = 80.634211442138
x16=−92.1533845053006x_{16} = -92.1533845053006
x17=−41.8879020478639x_{17} = -41.8879020478639
x18=−48.1710873550435x_{18} = -48.1710873550435
x19=15.707963267949x_{19} = 15.707963267949
x20=10.471975511966x_{20} = 10.471975511966
x21=−19.8967534727354x_{21} = -19.8967534727354
x22=−90.0589894029074x_{22} = -90.0589894029074
x23=32.4631240870945x_{23} = 32.4631240870945
x24=−21.9911485751286x_{24} = -21.9911485751286
x25=6.28318530717959x_{25} = 6.28318530717959
x26=−6.28318530717959x_{26} = -6.28318530717959
x27=26.1799387799149x_{27} = 26.1799387799149
x28=34.5575191894877x_{28} = 34.5575191894877
x29=−94.2477796076938x_{29} = -94.2477796076938
x30=−61.7846555205993x_{30} = -61.7846555205993
x31=746.651854003174x_{31} = 746.651854003174
x32=−17.8023583703422x_{32} = -17.8023583703422
x33=8.37758040957278x_{33} = 8.37758040957278
x34=46.0766922526503x_{34} = 46.0766922526503
x35=−33.5103216382911x_{35} = -33.5103216382911
x36=98.4365698124802x_{36} = 98.4365698124802
x37=28.2743338823081x_{37} = 28.2743338823081
x38=−98.4365698124802x_{38} = -98.4365698124802
x39=−31.4159265358979x_{39} = -31.4159265358979
x40=−79.5870138909414x_{40} = -79.5870138909414
x41=59.6902604182061x_{41} = 59.6902604182061
x42=109.955742875643x_{42} = 109.955742875643
x43=−81.6814089933346x_{43} = -81.6814089933346
x44=−85.870199198121x_{44} = -85.870199198121
x45=−37.6991118430775x_{45} = -37.6991118430775
x46=96.342174710087x_{46} = 96.342174710087
x47=−70.162235930172x_{47} = -70.162235930172
x48=−50.2654824574367x_{48} = -50.2654824574367
x49=−68.0678408277789x_{49} = -68.0678408277789
x50=94.2477796076938x_{50} = 94.2477796076938
x51=63.8790506229925x_{51} = 63.8790506229925
x52=−46.0766922526503x_{52} = -46.0766922526503
x53=−83.7758040957278x_{53} = -83.7758040957278
x54=30.3687289847013x_{54} = 30.3687289847013
x55=2.0943951023932x_{55} = 2.0943951023932
x56=56.5486677646163x_{56} = 56.5486677646163
x57=87.9645943005142x_{57} = 87.9645943005142
x58=65.9734457253857x_{58} = 65.9734457253857
x59=43.9822971502571x_{59} = 43.9822971502571
x60=−54.4542726622231x_{60} = -54.4542726622231
x61=19.8967534727354x_{61} = 19.8967534727354
x62=−2.0943951023932x_{62} = -2.0943951023932
x63=−10.471975511966x_{63} = -10.471975511966
x64=92.1533845053006x_{64} = 92.1533845053006
x65=90.0589894029074x_{65} = 90.0589894029074
x66=85.870199198121x_{66} = 85.870199198121
x67=−15.707963267949x_{67} = -15.707963267949
x68=68.0678408277789x_{68} = 68.0678408277789
x69=−35.6047167406843x_{69} = -35.6047167406843
x70=−72.2566310325652x_{70} = -72.2566310325652
x71=61.7846555205993x_{71} = 61.7846555205993
x72=0x_{72} = 0
x73=78.5398163397448x_{73} = 78.5398163397448
x74=4.18879020478639x_{74} = 4.18879020478639
x75=52.3598775598299x_{75} = 52.3598775598299
x76=24.0855436775217x_{76} = 24.0855436775217
x77=−24.0855436775217x_{77} = -24.0855436775217
x78=−87.9645943005142x_{78} = -87.9645943005142
x79=−43.9822971502571x_{79} = -43.9822971502571
x80=−8.37758040957278x_{80} = -8.37758040957278
x81=70.162235930172x_{81} = 70.162235930172
x82=21.9911485751286x_{82} = 21.9911485751286
x83=569.675467850949x_{83} = 569.675467850949
x84=39.7935069454707x_{84} = 39.7935069454707
x85=81.6814089933346x_{85} = 81.6814089933346
x86=−28.2743338823081x_{86} = -28.2743338823081
x87=−65.9734457253857x_{87} = -65.9734457253857
x88=48.1710873550435x_{88} = 48.1710873550435
x89=50.2654824574367x_{89} = 50.2654824574367
x90=−26.1799387799149x_{90} = -26.1799387799149
x91=100.530964914873x_{91} = 100.530964914873
x92=−11.5191730631626x_{92} = -11.5191730631626
x93=−99.4837673636768x_{93} = -99.4837673636768
x94=83.7758040957278x_{94} = 83.7758040957278
x95=−77.4926187885482x_{95} = -77.4926187885482
x96=76.4454212373516x_{96} = 76.4454212373516
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(3*x).
sin⁡(3⋅0)\sin{\left(3 \cdot 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
3cos⁡(3x)=03 \cos{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
 pi    
(--, 1)
 6     

 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=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=π6x_{1} = \frac{\pi}{6}
Decreasing at intervals
(−∞,π6]∪[π2,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
[π6,π2]\left[\frac{\pi}{6}, \frac{\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
−9sin⁡(3x)=0- 9 \sin{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π3x_{2} = \frac{\pi}{3}

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]∪[π3,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right)
Convex at the intervals
[0,π3]\left[0, \frac{\pi}{3}\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⁡(3x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(3 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⁡(3x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(3 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(3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(3 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⁡(3x)=−sin⁡(3x)\sin{\left(3 x \right)} = - \sin{\left(3 x \right)}
- No
sin⁡(3x)=sin⁡(3x)\sin{\left(3 x \right)} = \sin{\left(3 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = sin(3x)
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