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sin(2x+3)
Plotting a function graph/ sin(2x+3)

Graphing y = sin(2x+3)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(2*x + 3)
f(x)=sin⁡(2x+3)f{\left(x \right)} = \sin{\left(2 x + 3 \right)} 
f = sin(2*x + 3)
The graph of the function
0-40-30-20-1010203040502-2
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⁡(2x+3)=0\sin{\left(2 x + 3 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−32x_{1} = - \frac{3}{2}
x2=−32+π2x_{2} = - \frac{3}{2} + \frac{\pi}{2}
Numerical solution
x1=−32.9159265358979x_{1} = -32.9159265358979
x2=−43.9115008234622x_{2} = -43.9115008234622
x3=53.4778714378214x_{3} = 53.4778714378214
x4=67.6150383789755x_{4} = 67.6150383789755
x5=−15.6371669411541x_{5} = -15.6371669411541
x6=89.606186954104x_{6} = 89.606186954104
x7=15.7787595947439x_{7} = 15.7787595947439
x8=−67.4734457253857x_{8} = -67.4734457253857
x9=−20.3495559215388x_{9} = -20.3495559215388
x10=−29.7743338823081x_{10} = -29.7743338823081
x11=81.7522053201295x_{11} = 81.7522053201295
x12=37.7699081698724x_{12} = 37.7699081698724
x13=7.92477796076938x_{13} = 7.92477796076938
x14=−95.7477796076938x_{14} = -95.7477796076938
x15=70.7566310325652x_{15} = 70.7566310325652
x16=−58.0486677646163x_{16} = -58.0486677646163
x17=−51.7654824574367x_{17} = -51.7654824574367
x18=−37.6283155162826x_{18} = -37.6283155162826
x19=80.1814089933346x_{19} = 80.1814089933346
x20=−94.1769832808989x_{20} = -94.1769832808989
x21=−45.4822971502571x_{21} = -45.4822971502571
x22=78.6106126665397x_{22} = 78.6106126665397
x23=−73.7566310325652x_{23} = -73.7566310325652
x24=20.4911485751286x_{24} = 20.4911485751286
x25=−31.345130209103x_{25} = -31.345130209103
x26=28.345130209103x_{26} = 28.345130209103
x27=−56.4778714378214x_{27} = -56.4778714378214
x28=58.1902604182061x_{28} = 58.1902604182061
x29=94.3185759344887x_{29} = 94.3185759344887
x30=92.7477796076938x_{30} = 92.7477796076938
x31=−36.0575191894877x_{31} = -36.0575191894877
x32=51.9070751110265x_{32} = 51.9070751110265
x33=−86.3230016469244x_{33} = -86.3230016469244
x34=100.601761241668x_{34} = 100.601761241668
x35=86.4645943005142x_{35} = 86.4645943005142
x36=−81.6106126665397x_{36} = -81.6106126665397
x37=73.898223686155x_{37} = 73.898223686155
x38=−23.4911485751286x_{38} = -23.4911485751286
x39=−59.6194640914112x_{39} = -59.6194640914112
x40=−7.78318530717959x_{40} = -7.78318530717959
x41=59.761056745001x_{41} = 59.761056745001
x42=−6.21238898038469x_{42} = -6.21238898038469
x43=44.053093477052x_{43} = 44.053093477052
x44=1.64159265358979x_{44} = 1.64159265358979
x45=−100.460168588078x_{45} = -100.460168588078
x46=−28.2035375555132x_{46} = -28.2035375555132
x47=−53.3362787842316x_{47} = -53.3362787842316
x48=0.0707963267948966x_{48} = 0.0707963267948966
x49=22.0619449019235x_{49} = 22.0619449019235
x50=−34.4867228626928x_{50} = -34.4867228626928
x51=23.6327412287183x_{51} = 23.6327412287183
x52=−80.0398163397448x_{52} = -80.0398163397448
x53=50.3362787842316x_{53} = 50.3362787842316
x54=−12.4955742875643x_{54} = -12.4955742875643
x55=−9.35398163397448x_{55} = -9.35398163397448
x56=−21.9203522483337x_{56} = -21.9203522483337
x57=−50.1946861306418x_{57} = -50.1946861306418
x58=66.0442420521806x_{58} = 66.0442420521806
x59=45.6238898038469x_{59} = 45.6238898038469
x60=9.49557428756428x_{60} = 9.49557428756428
x61=−75.3274273593601x_{61} = -75.3274273593601
x62=−87.8937979737193x_{62} = -87.8937979737193
x63=31.4867228626928x_{63} = 31.4867228626928
x64=−89.4645943005142x_{64} = -89.4645943005142
x65=95.8893722612836x_{65} = 95.8893722612836
x66=34.6283155162826x_{66} = 34.6283155162826
x67=64.4734457253857x_{67} = 64.4734457253857
x68=−72.1858347057703x_{68} = -72.1858347057703
x69=88.0353906273091x_{69} = 88.0353906273091
x70=29.9159265358979x_{70} = 29.9159265358979
x71=−78.4690200129499x_{71} = -78.4690200129499
x72=36.1991118430775x_{72} = 36.1991118430775
x73=12.6371669411541x_{73} = 12.6371669411541
x74=−1.5x_{74} = -1.5
x75=42.4822971502571x_{75} = 42.4822971502571
x76=14.207963267949x_{76} = 14.207963267949
x77=6.35398163397448x_{77} = 6.35398163397448
x78=−65.9026493985908x_{78} = -65.9026493985908
x79=−64.3318530717959x_{79} = -64.3318530717959
x80=56.6194640914112x_{80} = 56.6194640914112
x81=−42.3407044966673x_{81} = -42.3407044966673
x82=−97.3185759344887x_{82} = -97.3185759344887
x83=48.7654824574367x_{83} = 48.7654824574367
x84=72.3274273593601x_{84} = 72.3274273593601
x85=−14.0663706143592x_{85} = -14.0663706143592
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2*x + 3).
sin⁡(2⋅0+3)\sin{\left(2 \cdot 0 + 3 \right)}
The result:
f(0)=sin⁡(3)f{\left(0 \right)} = \sin{\left(3 \right)}
The point:
(0, sin(3))
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
2cos⁡(2x+3)=02 \cos{\left(2 x + 3 \right)} = 0
Solve this equation
The roots of this equation
x1=−32+π4x_{1} = - \frac{3}{2} + \frac{\pi}{4}
x2=−32+3π4x_{2} = - \frac{3}{2} + \frac{3 \pi}{4}
The values of the extrema at the points:
   3   pi    
(- - + --, 1)
   2   4     

   3   3*pi     
(- - + ----, -1)
   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=−32+3π4x_{1} = - \frac{3}{2} + \frac{3 \pi}{4}
Maxima of the function at points:
x1=−32+π4x_{1} = - \frac{3}{2} + \frac{\pi}{4}
Decreasing at intervals
(−∞,−32+π4]∪[−32+3π4,∞)\left(-\infty, - \frac{3}{2} + \frac{\pi}{4}\right] \cup \left[- \frac{3}{2} + \frac{3 \pi}{4}, \infty\right)
Increasing at intervals
[−32+π4,−32+3π4]\left[- \frac{3}{2} + \frac{\pi}{4}, - \frac{3}{2} + \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
−4sin⁡(2x+3)=0- 4 \sin{\left(2 x + 3 \right)} = 0
Solve this equation
The roots of this equation
x1=−32x_{1} = - \frac{3}{2}
x2=−32+π2x_{2} = - \frac{3}{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
(−∞,−32]∪[−32+π2,∞)\left(-\infty, - \frac{3}{2}\right] \cup \left[- \frac{3}{2} + \frac{\pi}{2}, \infty\right)
Convex at the intervals
[−32,−32+π2]\left[- \frac{3}{2}, - \frac{3}{2} + \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→−∞sin⁡(2x+3)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(2 x + 3 \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⁡(2x+3)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(2 x + 3 \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(2*x + 3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x+3)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x + 3 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x+3)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x + 3 \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⁡(2x+3)=−sin⁡(2x−3)\sin{\left(2 x + 3 \right)} = - \sin{\left(2 x - 3 \right)}
- No
sin⁡(2x+3)=sin⁡(2x−3)\sin{\left(2 x + 3 \right)} = \sin{\left(2 x - 3 \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(2x+3)
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