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Plotting a function graph/ sin4x

Graphing y = sin4x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(4*x)
f(x)=sin⁡(4x)f{\left(x \right)} = \sin{\left(4 x \right)} 
f = sin(4*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⁡(4x)=0\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=10.2101761241668x_{1} = 10.2101761241668
x2=51.8362787842316x_{2} = 51.8362787842316
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=68.329640215578x_{4} = 68.329640215578
x5=−91.8915851175014x_{5} = -91.8915851175014
x6=3.92699081698724x_{6} = 3.92699081698724
x7=36.1283155162826x_{7} = 36.1283155162826
x8=−69.1150383789755x_{8} = -69.1150383789755
x9=−3.92699081698724x_{9} = -3.92699081698724
x10=−11.7809724509617x_{10} = -11.7809724509617
x11=−59.6902604182061x_{11} = -59.6902604182061
x12=−54.1924732744239x_{12} = -54.1924732744239
x13=21.9911485751286x_{13} = 21.9911485751286
x14=43.9822971502571x_{14} = 43.9822971502571
x15=69.9004365423729x_{15} = 69.9004365423729
x16=98.174770424681x_{16} = 98.174770424681
x17=−76.1836218495525x_{17} = -76.1836218495525
x18=−25.9181393921158x_{18} = -25.9181393921158
x19=−80.1106126665397x_{19} = -80.1106126665397
x20=7.85398163397448x_{20} = 7.85398163397448
x21=−29.845130209103x_{21} = -29.845130209103
x22=−33.7721210260903x_{22} = -33.7721210260903
x23=−23.5619449019235x_{23} = -23.5619449019235
x24=14.1371669411541x_{24} = 14.1371669411541
x25=181.426975744811x_{25} = 181.426975744811
x26=−18.0641577581413x_{26} = -18.0641577581413
x27=58.1194640914112x_{27} = 58.1194640914112
x28=94.2477796076938x_{28} = 94.2477796076938
x29=−51.8362787842316x_{29} = -51.8362787842316
x30=−69.9004365423729x_{30} = -69.9004365423729
x31=80.1106126665397x_{31} = 80.1106126665397
x32=2.35619449019234x_{32} = 2.35619449019234
x33=−19.6349540849362x_{33} = -19.6349540849362
x34=24.3473430653209x_{34} = 24.3473430653209
x35=64.4026493985908x_{35} = 64.4026493985908
x36=−85.6083998103219x_{36} = -85.6083998103219
x37=−77.7544181763474x_{37} = -77.7544181763474
x38=0x_{38} = 0
x39=90.3207887907066x_{39} = 90.3207887907066
x40=−58.1194640914112x_{40} = -58.1194640914112
x41=−81.6814089933346x_{41} = -81.6814089933346
x42=86.3937979737193x_{42} = 86.3937979737193
x43=40.0553063332699x_{43} = 40.0553063332699
x44=−64.4026493985908x_{44} = -64.4026493985908
x45=10.9955742875643x_{45} = 10.9955742875643
x46=−14.1371669411541x_{46} = -14.1371669411541
x47=−87.9645943005142x_{47} = -87.9645943005142
x48=6.28318530717959x_{48} = 6.28318530717959
x49=−98.174770424681x_{49} = -98.174770424681
x50=−62.0464549083984x_{50} = -62.0464549083984
x51=−37.6991118430775x_{51} = -37.6991118430775
x52=−7.85398163397448x_{52} = -7.85398163397448
x53=84.037603483527x_{53} = 84.037603483527
x54=−63.6172512351933x_{54} = -63.6172512351933
x55=54.1924732744239x_{55} = 54.1924732744239
x56=65.9734457253857x_{56} = 65.9734457253857
x57=−99.7455667514759x_{57} = -99.7455667514759
x58=20.4203522483337x_{58} = 20.4203522483337
x59=46.3384916404494x_{59} = 46.3384916404494
x60=−32.2013246992954x_{60} = -32.2013246992954
x61=72.2566310325652x_{61} = 72.2566310325652
x62=−43.9822971502571x_{62} = -43.9822971502571
x63=−41.6261026600648x_{63} = -41.6261026600648
x64=73.8274273593601x_{64} = 73.8274273593601
x65=−88.7499924639117x_{65} = -88.7499924639117
x66=47.9092879672443x_{66} = 47.9092879672443
x67=17.2787595947439x_{67} = 17.2787595947439
x68=18.0641577581413x_{68} = 18.0641577581413
x69=−55.7632696012188x_{69} = -55.7632696012188
x70=−10.9955742875643x_{70} = -10.9955742875643
x71=−47.9092879672443x_{71} = -47.9092879672443
x72=−95.8185759344887x_{72} = -95.8185759344887
x73=87.9645943005142x_{73} = 87.9645943005142
x74=42.4115008234622x_{74} = 42.4115008234622
x75=95.8185759344887x_{75} = 95.8185759344887
x76=62.0464549083984x_{76} = 62.0464549083984
x77=32.2013246992954x_{77} = 32.2013246992954
x78=29.845130209103x_{78} = 29.845130209103
x79=−109.170344712245x_{79} = -109.170344712245
x80=−73.8274273593601x_{80} = -73.8274273593601
x81=28.2743338823081x_{81} = 28.2743338823081
x82=83.2522053201295x_{82} = 83.2522053201295
x83=−84.037603483527x_{83} = -84.037603483527
x84=−21.9911485751286x_{84} = -21.9911485751286
x85=−65.9734457253857x_{85} = -65.9734457253857
x86=76.1836218495525x_{86} = 76.1836218495525
x87=−15.707963267949x_{87} = -15.707963267949
x88=−45.553093477052x_{88} = -45.553093477052
x89=−1.5707963267949x_{89} = -1.5707963267949
x90=25.9181393921158x_{90} = 25.9181393921158
x91=−40.0553063332699x_{91} = -40.0553063332699
x92=54.9778714378214x_{92} = 54.9778714378214
x93=50.2654824574367x_{93} = 50.2654824574367
x94=91.8915851175014x_{94} = 91.8915851175014
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(4*x).
sin⁡(0⋅4)\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
4cos⁡(4x)=04 \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    
(--, 1)
 8     

 3*pi     
(----, -1)
  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
−16sin⁡(4x)=0- 16 \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→−∞sin⁡(4x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(4 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⁡(4x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(4 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(4*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{\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→∞(sin⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{\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:
sin⁡(4x)=−sin⁡(4x)\sin{\left(4 x \right)} = - \sin{\left(4 x \right)}
- No
sin⁡(4x)=sin⁡(4x)\sin{\left(4 x \right)} = \sin{\left(4 x \right)}
- Yes
so, the function
is
odd
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