Mister Exam

Other calculators

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

Graphing y = sin(2*x+3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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