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Plotting a function graph/ -2sin2x

Graphing y = -2sin2x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=12.5663706143592x_{1} = 12.5663706143592
x2=78.5398163397448x_{2} = 78.5398163397448
x3=−65.9734457253857x_{3} = -65.9734457253857
x4=−15.707963267949x_{4} = -15.707963267949
x5=86.3937979737193x_{5} = 86.3937979737193
x6=−48.6946861306418x_{6} = -48.6946861306418
x7=50.2654824574367x_{7} = 50.2654824574367
x8=81.6814089933346x_{8} = 81.6814089933346
x9=−64.4026493985908x_{9} = -64.4026493985908
x10=42.4115008234622x_{10} = 42.4115008234622
x11=73.8274273593601x_{11} = 73.8274273593601
x12=45.553093477052x_{12} = 45.553093477052
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−75.398223686155x_{14} = -75.398223686155
x15=590.619418874881x_{15} = 590.619418874881
x16=−1.5707963267949x_{16} = -1.5707963267949
x17=−58.1194640914112x_{17} = -58.1194640914112
x18=56.5486677646163x_{18} = 56.5486677646163
x19=−61.261056745001x_{19} = -61.261056745001
x20=−51.8362787842316x_{20} = -51.8362787842316
x21=15.707963267949x_{21} = 15.707963267949
x22=7.85398163397448x_{22} = 7.85398163397448
x23=−86.3937979737193x_{23} = -86.3937979737193
x24=58.1194640914112x_{24} = 58.1194640914112
x25=23.5619449019235x_{25} = 23.5619449019235
x26=−67.5442420521806x_{26} = -67.5442420521806
x27=−59.6902604182061x_{27} = -59.6902604182061
x28=21.9911485751286x_{28} = 21.9911485751286
x29=6.28318530717959x_{29} = 6.28318530717959
x30=−119.380520836412x_{30} = -119.380520836412
x31=−87.9645943005142x_{31} = -87.9645943005142
x32=−28.2743338823081x_{32} = -28.2743338823081
x33=−9.42477796076938x_{33} = -9.42477796076938
x34=−95.8185759344887x_{34} = -95.8185759344887
x35=59.6902604182061x_{35} = 59.6902604182061
x36=29.845130209103x_{36} = 29.845130209103
x37=28.2743338823081x_{37} = 28.2743338823081
x38=80.1106126665397x_{38} = 80.1106126665397
x39=31.4159265358979x_{39} = 31.4159265358979
x40=94.2477796076938x_{40} = 94.2477796076938
x41=−72.2566310325652x_{41} = -72.2566310325652
x42=−80.1106126665397x_{42} = -80.1106126665397
x43=51.8362787842316x_{43} = 51.8362787842316
x44=−29.845130209103x_{44} = -29.845130209103
x45=−97.3893722612836x_{45} = -97.3893722612836
x46=37.6991118430775x_{46} = 37.6991118430775
x47=−483.805268652828x_{47} = -483.805268652828
x48=−20.4203522483337x_{48} = -20.4203522483337
x49=−50.2654824574367x_{49} = -50.2654824574367
x50=−94.2477796076938x_{50} = -94.2477796076938
x51=−17.2787595947439x_{51} = -17.2787595947439
x52=−40.8407044966673x_{52} = -40.8407044966673
x53=20.4203522483337x_{53} = 20.4203522483337
x54=67.5442420521806x_{54} = 67.5442420521806
x55=14.1371669411541x_{55} = 14.1371669411541
x56=4.71238898038469x_{56} = 4.71238898038469
x57=−37.6991118430775x_{57} = -37.6991118430775
x58=70.6858347057703x_{58} = 70.6858347057703
x59=−23.5619449019235x_{59} = -23.5619449019235
x60=−83.2522053201295x_{60} = -83.2522053201295
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=1.5707963267949x_{62} = 1.5707963267949
x63=36.1283155162826x_{63} = 36.1283155162826
x64=−81.6814089933346x_{64} = -81.6814089933346
x65=43.9822971502571x_{65} = 43.9822971502571
x66=95.8185759344887x_{66} = 95.8185759344887
x67=−14.1371669411541x_{67} = -14.1371669411541
x68=−31.4159265358979x_{68} = -31.4159265358979
x69=0x_{69} = 0
x70=−21.9911485751286x_{70} = -21.9911485751286
x71=−39.2699081698724x_{71} = -39.2699081698724
x72=26.7035375555132x_{72} = 26.7035375555132
x73=−89.5353906273091x_{73} = -89.5353906273091
x74=100.530964914873x_{74} = 100.530964914873
x75=34.5575191894877x_{75} = 34.5575191894877
x76=48.6946861306418x_{76} = 48.6946861306418
x77=65.9734457253857x_{77} = 65.9734457253857
x78=−45.553093477052x_{78} = -45.553093477052
x79=113.097335529233x_{79} = 113.097335529233
x80=−53.4070751110265x_{80} = -53.4070751110265
x81=−7.85398163397448x_{81} = -7.85398163397448
x82=64.4026493985908x_{82} = 64.4026493985908
x83=−6.28318530717959x_{83} = -6.28318530717959
x84=−73.8274273593601x_{84} = -73.8274273593601
x85=−43.9822971502571x_{85} = -43.9822971502571
x86=92.6769832808989x_{86} = 92.6769832808989
x87=72.2566310325652x_{87} = 72.2566310325652
x88=−42.4115008234622x_{88} = -42.4115008234622
x89=87.9645943005142x_{89} = 87.9645943005142
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -2*sin(2*x).
−2sin⁡(0⋅2)- 2 \sin{\left(0 \cdot 2 \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⁡(2x)=0- 4 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
The values of the extrema at the points:
 pi     
(--, -2)
 4      

 3*pi    
(----, 2)
  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=3π4x_{1} = \frac{3 \pi}{4}
Decreasing at intervals
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]
Increasing at intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\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
8sin⁡(2x)=08 \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}

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,π2]\left[0, \frac{\pi}{2}\right]
Convex at the intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\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⁡(2x))=⟨−2,2⟩\lim_{x \to -\infty}\left(- 2 \sin{\left(2 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⁡(2x))=⟨−2,2⟩\lim_{x \to \infty}\left(- 2 \sin{\left(2 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(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−2sin⁡(2x)x)=0\lim_{x \to -\infty}\left(- \frac{2 \sin{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−2sin⁡(2x)x)=0\lim_{x \to \infty}\left(- \frac{2 \sin{\left(2 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⁡(2x)=2sin⁡(2x)- 2 \sin{\left(2 x \right)} = 2 \sin{\left(2 x \right)}
- No
−2sin⁡(2x)=−2sin⁡(2x)- 2 \sin{\left(2 x \right)} = - 2 \sin{\left(2 x \right)}
- Yes
so, the function
is
odd
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