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Plotting a function graph/ 3sin4x

Graphing y = 3sin4x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*sin(4*x)
f(x)=3sin⁡(4x)f{\left(x \right)} = 3 \sin{\left(4 x \right)} 
f = 3*sin(4*x)
The graph of the function
02468-8-6-4-2-10105-5
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:
3sin⁡(4x)=03 \sin{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π4x_{2} = \frac{\pi}{4}
Numerical solution
x1=25.9181393921158x_{1} = 25.9181393921158
x2=−15.707963267949x_{2} = -15.707963267949
x3=42.4115008234622x_{3} = 42.4115008234622
x4=21.9911485751286x_{4} = 21.9911485751286
x5=−99.7455667514759x_{5} = -99.7455667514759
x6=0x_{6} = 0
x7=−54.1924732744239x_{7} = -54.1924732744239
x8=−29.845130209103x_{8} = -29.845130209103
x9=17.2787595947439x_{9} = 17.2787595947439
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=18.0641577581413x_{11} = 18.0641577581413
x12=−36.1283155162826x_{12} = -36.1283155162826
x13=28.2743338823081x_{13} = 28.2743338823081
x14=86.3937979737193x_{14} = 86.3937979737193
x15=−10.9955742875643x_{15} = -10.9955742875643
x16=72.2566310325652x_{16} = 72.2566310325652
x17=46.3384916404494x_{17} = 46.3384916404494
x18=87.9645943005142x_{18} = 87.9645943005142
x19=−95.8185759344887x_{19} = -95.8185759344887
x20=−77.7544181763474x_{20} = -77.7544181763474
x21=−76.1836218495525x_{21} = -76.1836218495525
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=50.2654824574367x_{23} = 50.2654824574367
x24=−14.1371669411541x_{24} = -14.1371669411541
x25=76.1836218495525x_{25} = 76.1836218495525
x26=54.1924732744239x_{26} = 54.1924732744239
x27=58.1194640914112x_{27} = 58.1194640914112
x28=10.9955742875643x_{28} = 10.9955742875643
x29=2.35619449019234x_{29} = 2.35619449019234
x30=84.037603483527x_{30} = 84.037603483527
x31=−62.0464549083984x_{31} = -62.0464549083984
x32=91.8915851175014x_{32} = 91.8915851175014
x33=−85.6083998103219x_{33} = -85.6083998103219
x34=−33.7721210260903x_{34} = -33.7721210260903
x35=−23.5619449019235x_{35} = -23.5619449019235
x36=10.2101761241668x_{36} = 10.2101761241668
x37=−81.6814089933346x_{37} = -81.6814089933346
x38=94.2477796076938x_{38} = 94.2477796076938
x39=−19.6349540849362x_{39} = -19.6349540849362
x40=−80.1106126665397x_{40} = -80.1106126665397
x41=−1.5707963267949x_{41} = -1.5707963267949
x42=36.1283155162826x_{42} = 36.1283155162826
x43=188.495559215388x_{43} = 188.495559215388
x44=−69.9004365423729x_{44} = -69.9004365423729
x45=−32.2013246992954x_{45} = -32.2013246992954
x46=−40.0553063332699x_{46} = -40.0553063332699
x47=83.2522053201295x_{47} = 83.2522053201295
x48=−45.553093477052x_{48} = -45.553093477052
x49=65.9734457253857x_{49} = 65.9734457253857
x50=73.8274273593601x_{50} = 73.8274273593601
x51=98.174770424681x_{51} = 98.174770424681
x52=90.3207887907066x_{52} = 90.3207887907066
x53=−25.9181393921158x_{53} = -25.9181393921158
x54=20.4203522483337x_{54} = 20.4203522483337
x55=68.329640215578x_{55} = 68.329640215578
x56=−87.9645943005142x_{56} = -87.9645943005142
x57=3.92699081698724x_{57} = 3.92699081698724
x58=−98.174770424681x_{58} = -98.174770424681
x59=−41.6261026600648x_{59} = -41.6261026600648
x60=95.8185759344887x_{60} = 95.8185759344887
x61=−47.9092879672443x_{61} = -47.9092879672443
x62=−58.1194640914112x_{62} = -58.1194640914112
x63=−55.7632696012188x_{63} = -55.7632696012188
x64=24.3473430653209x_{64} = 24.3473430653209
x65=−91.8915851175014x_{65} = -91.8915851175014
x66=80.1106126665397x_{66} = 80.1106126665397
x67=−84.037603483527x_{67} = -84.037603483527
x68=40.0553063332699x_{68} = 40.0553063332699
x69=−109.170344712245x_{69} = -109.170344712245
x70=7.85398163397448x_{70} = 7.85398163397448
x71=−69.1150383789755x_{71} = -69.1150383789755
x72=29.845130209103x_{72} = 29.845130209103
x73=−65.9734457253857x_{73} = -65.9734457253857
x74=−11.7809724509617x_{74} = -11.7809724509617
x75=−63.6172512351933x_{75} = -63.6172512351933
x76=43.9822971502571x_{76} = 43.9822971502571
x77=14.1371669411541x_{77} = 14.1371669411541
x78=−37.6991118430775x_{78} = -37.6991118430775
x79=−59.6902604182061x_{79} = -59.6902604182061
x80=−64.4026493985908x_{80} = -64.4026493985908
x81=51.8362787842316x_{81} = 51.8362787842316
x82=−88.7499924639117x_{82} = -88.7499924639117
x83=64.4026493985908x_{83} = 64.4026493985908
x84=−73.8274273593601x_{84} = -73.8274273593601
x85=6.28318530717959x_{85} = 6.28318530717959
x86=−3.92699081698724x_{86} = -3.92699081698724
x87=32.2013246992954x_{87} = 32.2013246992954
x88=−51.8362787842316x_{88} = -51.8362787842316
x89=−7.85398163397448x_{89} = -7.85398163397448
x90=47.9092879672443x_{90} = 47.9092879672443
x91=69.9004365423729x_{91} = 69.9004365423729
x92=62.0464549083984x_{92} = 62.0464549083984
x93=54.9778714378214x_{93} = 54.9778714378214
x94=614.181363776805x_{94} = 614.181363776805
x95=−18.0641577581413x_{95} = -18.0641577581413
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*sin(4*x).
3sin⁡(0⋅4)3 \sin{\left(0 \cdot 4 \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
12cos⁡(4x)=012 \cos{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=π8x_{1} = \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}
The values of the extrema at the points:
 pi    
(--, 3)
 8     

 3*pi     
(----, -3)
  8


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=3π8x_{1} = \frac{3 \pi}{8}
Maxima of the function at points:
x1=π8x_{1} = \frac{\pi}{8}
Decreasing at intervals
(−∞,π8]∪[3π8,∞)\left(-\infty, \frac{\pi}{8}\right] \cup \left[\frac{3 \pi}{8}, \infty\right)
Increasing at intervals
[π8,3π8]\left[\frac{\pi}{8}, \frac{3 \pi}{8}\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
−48sin⁡(4x)=0- 48 \sin{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π4x_{2} = \frac{\pi}{4}

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]∪[π4,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{4}, \infty\right)
Convex at the intervals
[0,π4]\left[0, \frac{\pi}{4}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3sin⁡(4x))=⟨−3,3⟩\lim_{x \to -\infty}\left(3 \sin{\left(4 x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(3sin⁡(4x))=⟨−3,3⟩\lim_{x \to \infty}\left(3 \sin{\left(4 x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*sin(4*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(4 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(4 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:
3sin⁡(4x)=−3sin⁡(4x)3 \sin{\left(4 x \right)} = - 3 \sin{\left(4 x \right)}
- No
3sin⁡(4x)=3sin⁡(4x)3 \sin{\left(4 x \right)} = 3 \sin{\left(4 x \right)}
- Yes
so, the function
is
odd
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