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2sin4x
Plotting a function graph/ 2sin4x

Graphing y = 2sin4x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(4*x)
f(x)=2sin⁡(4x)f{\left(x \right)} = 2 \sin{\left(4 x \right)} 
f = 2*sin(4*x)
The graph of the function
0102030405060708090-105-5
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:
2sin⁡(4x)=02 \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=83.2522053201295x_{1} = 83.2522053201295
x2=95.8185759344887x_{2} = 95.8185759344887
x3=76.1836218495525x_{3} = 76.1836218495525
x4=−19.6349540849362x_{4} = -19.6349540849362
x5=91.8915851175014x_{5} = 91.8915851175014
x6=17.2787595947439x_{6} = 17.2787595947439
x7=32.2013246992954x_{7} = 32.2013246992954
x8=72.2566310325652x_{8} = 72.2566310325652
x9=−59.6902604182061x_{9} = -59.6902604182061
x10=68.329640215578x_{10} = 68.329640215578
x11=42.4115008234622x_{11} = 42.4115008234622
x12=−7.85398163397448x_{12} = -7.85398163397448
x13=−91.8915851175014x_{13} = -91.8915851175014
x14=25.9181393921158x_{14} = 25.9181393921158
x15=51.8362787842316x_{15} = 51.8362787842316
x16=69.9004365423729x_{16} = 69.9004365423729
x17=10.2101761241668x_{17} = 10.2101761241668
x18=−41.6261026600648x_{18} = -41.6261026600648
x19=−36.1283155162826x_{19} = -36.1283155162826
x20=20.4203522483337x_{20} = 20.4203522483337
x21=−23.5619449019235x_{21} = -23.5619449019235
x22=−76.1836218495525x_{22} = -76.1836218495525
x23=36.1283155162826x_{23} = 36.1283155162826
x24=−47.9092879672443x_{24} = -47.9092879672443
x25=3.92699081698724x_{25} = 3.92699081698724
x26=−21.9911485751286x_{26} = -21.9911485751286
x27=6.28318530717959x_{27} = 6.28318530717959
x28=−33.7721210260903x_{28} = -33.7721210260903
x29=−80.1106126665397x_{29} = -80.1106126665397
x30=86.3937979737193x_{30} = 86.3937979737193
x31=−11.7809724509617x_{31} = -11.7809724509617
x32=−3.92699081698724x_{32} = -3.92699081698724
x33=−98.174770424681x_{33} = -98.174770424681
x34=−14.1371669411541x_{34} = -14.1371669411541
x35=47.9092879672443x_{35} = 47.9092879672443
x36=98.174770424681x_{36} = 98.174770424681
x37=2.35619449019234x_{37} = 2.35619449019234
x38=−64.4026493985908x_{38} = -64.4026493985908
x39=28.2743338823081x_{39} = 28.2743338823081
x40=−10.9955742875643x_{40} = -10.9955742875643
x41=58.1194640914112x_{41} = 58.1194640914112
x42=−81.6814089933346x_{42} = -81.6814089933346
x43=−58.1194640914112x_{43} = -58.1194640914112
x44=−37.6991118430775x_{44} = -37.6991118430775
x45=−63.6172512351933x_{45} = -63.6172512351933
x46=73.8274273593601x_{46} = 73.8274273593601
x47=−18.0641577581413x_{47} = -18.0641577581413
x48=94.2477796076938x_{48} = 94.2477796076938
x49=−51.8362787842316x_{49} = -51.8362787842316
x50=−69.9004365423729x_{50} = -69.9004365423729
x51=87.9645943005142x_{51} = 87.9645943005142
x52=65.9734457253857x_{52} = 65.9734457253857
x53=10.9955742875643x_{53} = 10.9955742875643
x54=−55.7632696012188x_{54} = -55.7632696012188
x55=−25.9181393921158x_{55} = -25.9181393921158
x56=7.85398163397448x_{56} = 7.85398163397448
x57=54.9778714378214x_{57} = 54.9778714378214
x58=40.0553063332699x_{58} = 40.0553063332699
x59=43.9822971502571x_{59} = 43.9822971502571
x60=−40.0553063332699x_{60} = -40.0553063332699
x61=18.0641577581413x_{61} = 18.0641577581413
x62=−109.170344712245x_{62} = -109.170344712245
x63=−45.553093477052x_{63} = -45.553093477052
x64=90.3207887907066x_{64} = 90.3207887907066
x65=−15.707963267949x_{65} = -15.707963267949
x66=−69.1150383789755x_{66} = -69.1150383789755
x67=24.3473430653209x_{67} = 24.3473430653209
x68=46.3384916404494x_{68} = 46.3384916404494
x69=−73.8274273593601x_{69} = -73.8274273593601
x70=−77.7544181763474x_{70} = -77.7544181763474
x71=−32.2013246992954x_{71} = -32.2013246992954
x72=0x_{72} = 0
x73=−54.1924732744239x_{73} = -54.1924732744239
x74=64.4026493985908x_{74} = 64.4026493985908
x75=62.0464549083984x_{75} = 62.0464549083984
x76=80.1106126665397x_{76} = 80.1106126665397
x77=−85.6083998103219x_{77} = -85.6083998103219
x78=−87.9645943005142x_{78} = -87.9645943005142
x79=181.426975744811x_{79} = 181.426975744811
x80=−43.9822971502571x_{80} = -43.9822971502571
x81=−95.8185759344887x_{81} = -95.8185759344887
x82=14.1371669411541x_{82} = 14.1371669411541
x83=−88.7499924639117x_{83} = -88.7499924639117
x84=21.9911485751286x_{84} = 21.9911485751286
x85=84.037603483527x_{85} = 84.037603483527
x86=−84.037603483527x_{86} = -84.037603483527
x87=−65.9734457253857x_{87} = -65.9734457253857
x88=−1.5707963267949x_{88} = -1.5707963267949
x89=50.2654824574367x_{89} = 50.2654824574367
x90=54.1924732744239x_{90} = 54.1924732744239
x91=−29.845130209103x_{91} = -29.845130209103
x92=−99.7455667514759x_{92} = -99.7455667514759
x93=29.845130209103x_{93} = 29.845130209103
x94=−62.0464549083984x_{94} = -62.0464549083984
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(4*x).
2sin⁡(4⋅0)2 \sin{\left(4 \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
8cos⁡(4x)=08 \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    
(--, 2)
 8     

 3*pi     
(----, -2)
  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
−32sin⁡(4x)=0- 32 \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→−∞(2sin⁡(4x))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(4 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(2sin⁡(4x))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(4 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(4*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{2 \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→∞(2sin⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{2 \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:
2sin⁡(4x)=−2sin⁡(4x)2 \sin{\left(4 x \right)} = - 2 \sin{\left(4 x \right)}
- No
2sin⁡(4x)=2sin⁡(4x)2 \sin{\left(4 x \right)} = 2 \sin{\left(4 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = 2sin4x
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