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Plotting a function graph/ y=1,5sin2x

Graphing y = y=1,5sin2x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       3*sin(2*x)
f(x) = ----------
           2
f(x)=3sin⁡(2x)2f{\left(x \right)} = \frac{3 \sin{\left(2 x \right)}}{2} 
f = 3*sin(2*x)/2
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⁡(2x)2=0\frac{3 \sin{\left(2 x \right)}}{2} = 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=−97.3893722612836x_{1} = -97.3893722612836
x2=−43.9822971502571x_{2} = -43.9822971502571
x3=−29.845130209103x_{3} = -29.845130209103
x4=56.5486677646163x_{4} = 56.5486677646163
x5=28.2743338823081x_{5} = 28.2743338823081
x6=29.845130209103x_{6} = 29.845130209103
x7=−81.6814089933346x_{7} = -81.6814089933346
x8=−15.707963267949x_{8} = -15.707963267949
x9=72.2566310325652x_{9} = 72.2566310325652
x10=−39.2699081698724x_{10} = -39.2699081698724
x11=95.8185759344887x_{11} = 95.8185759344887
x12=−9787.631912259x_{12} = -9787.631912259
x13=45.553093477052x_{13} = 45.553093477052
x14=−53.4070751110265x_{14} = -53.4070751110265
x15=100.530964914873x_{15} = 100.530964914873
x16=−9.42477796076938x_{16} = -9.42477796076938
x17=81.6814089933346x_{17} = 81.6814089933346
x18=70.6858347057703x_{18} = 70.6858347057703
x19=−21.9911485751286x_{19} = -21.9911485751286
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=−119.380520836412x_{21} = -119.380520836412
x22=42.4115008234622x_{22} = 42.4115008234622
x23=−72.2566310325652x_{23} = -72.2566310325652
x24=15.707963267949x_{24} = 15.707963267949
x25=50.2654824574367x_{25} = 50.2654824574367
x26=−28.2743338823081x_{26} = -28.2743338823081
x27=−6.28318530717959x_{27} = -6.28318530717959
x28=−17.2787595947439x_{28} = -17.2787595947439
x29=−48.6946861306418x_{29} = -48.6946861306418
x30=590.619418874881x_{30} = 590.619418874881
x31=−80.1106126665397x_{31} = -80.1106126665397
x32=113.097335529233x_{32} = 113.097335529233
x33=−87.9645943005142x_{33} = -87.9645943005142
x34=48.6946861306418x_{34} = 48.6946861306418
x35=−20.4203522483337x_{35} = -20.4203522483337
x36=0x_{36} = 0
x37=20.4203522483337x_{37} = 20.4203522483337
x38=89.5353906273091x_{38} = 89.5353906273091
x39=31.4159265358979x_{39} = 31.4159265358979
x40=−65.9734457253857x_{40} = -65.9734457253857
x41=−73.8274273593601x_{41} = -73.8274273593601
x42=6.28318530717959x_{42} = 6.28318530717959
x43=−67.5442420521806x_{43} = -67.5442420521806
x44=−36.1283155162826x_{44} = -36.1283155162826
x45=94.2477796076938x_{45} = 94.2477796076938
x46=−61.261056745001x_{46} = -61.261056745001
x47=65.9734457253857x_{47} = 65.9734457253857
x48=36.1283155162826x_{48} = 36.1283155162826
x49=26.7035375555132x_{49} = 26.7035375555132
x50=21.9911485751286x_{50} = 21.9911485751286
x51=78.5398163397448x_{51} = 78.5398163397448
x52=−89.5353906273091x_{52} = -89.5353906273091
x53=4.71238898038469x_{53} = 4.71238898038469
x54=7.85398163397448x_{54} = 7.85398163397448
x55=14.1371669411541x_{55} = 14.1371669411541
x56=86.3937979737193x_{56} = 86.3937979737193
x57=87.9645943005142x_{57} = 87.9645943005142
x58=12.5663706143592x_{58} = 12.5663706143592
x59=51.8362787842316x_{59} = 51.8362787842316
x60=−37.6991118430775x_{60} = -37.6991118430775
x61=−51.8362787842316x_{61} = -51.8362787842316
x62=−1.5707963267949x_{62} = -1.5707963267949
x63=59.6902604182061x_{63} = 59.6902604182061
x64=−23.5619449019235x_{64} = -23.5619449019235
x65=64.4026493985908x_{65} = 64.4026493985908
x66=43.9822971502571x_{66} = 43.9822971502571
x67=−50.2654824574367x_{67} = -50.2654824574367
x68=37.6991118430775x_{68} = 37.6991118430775
x69=−95.8185759344887x_{69} = -95.8185759344887
x70=−64.4026493985908x_{70} = -64.4026493985908
x71=80.1106126665397x_{71} = 80.1106126665397
x72=−40.8407044966673x_{72} = -40.8407044966673
x73=73.8274273593601x_{73} = 73.8274273593601
x74=−59.6902604182061x_{74} = -59.6902604182061
x75=−7.85398163397448x_{75} = -7.85398163397448
x76=−75.398223686155x_{76} = -75.398223686155
x77=−86.3937979737193x_{77} = -86.3937979737193
x78=23.5619449019235x_{78} = 23.5619449019235
x79=92.6769832808989x_{79} = 92.6769832808989
x80=−45.553093477052x_{80} = -45.553093477052
x81=−83.2522053201295x_{81} = -83.2522053201295
x82=1.5707963267949x_{82} = 1.5707963267949
x83=−58.1194640914112x_{83} = -58.1194640914112
x84=58.1194640914112x_{84} = 58.1194640914112
x85=−94.2477796076938x_{85} = -94.2477796076938
x86=−42.4115008234622x_{86} = -42.4115008234622
x87=67.5442420521806x_{87} = 67.5442420521806
x88=34.5575191894877x_{88} = 34.5575191894877
x89=−31.4159265358979x_{89} = -31.4159265358979
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*sin(2*x)/2.
3sin⁡(0⋅2)2\frac{3 \sin{\left(0 \cdot 2 \right)}}{2}
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⁡(2x)=03 \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      
(--, 3/2)
 4       

 3*pi       
(----, -3/2)
  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=3π4x_{1} = \frac{3 \pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Increasing at intervals
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\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
−6sin⁡(2x)=0- 6 \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(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
[0,π2]\left[0, \frac{\pi}{2}\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⁡(2x)2)=⟨−32,32⟩\lim_{x \to -\infty}\left(\frac{3 \sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
lim⁡x→∞(3sin⁡(2x)2)=⟨−32,32⟩\lim_{x \to \infty}\left(\frac{3 \sin{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*sin(2*x)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(2x)2x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(2 x \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(2x)2x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(2 x \right)}}{2 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⁡(2x)2=−3sin⁡(2x)2\frac{3 \sin{\left(2 x \right)}}{2} = - \frac{3 \sin{\left(2 x \right)}}{2}
- No
3sin⁡(2x)2=3sin⁡(2x)2\frac{3 \sin{\left(2 x \right)}}{2} = \frac{3 \sin{\left(2 x \right)}}{2}
- Yes
so, the function
is
odd
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