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  • Graphing y =:
  • -x^2-2x+15
  • (x^2+3)/(x-1) (x^2+3)/(x-1)
  • (9x^2-1)/x
  • 6x-x^2

  • Identical expressions

  • sin(two *x+ three)^(two)
  • sinus of (2 multiply by x plus 3) to the power of (2)
  • sinus of (two multiply by x plus three) to the power of (two)
  • sin(2*x+3)(2)
  • sin2*x+32
  • sin(2x+3)^(2)
  • sin(2x+3)(2)
  • sin2x+32
  • sin2x+3^2

  • Similar expressions

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

Graphing y = sin(2*x+3)^(2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2         
f(x) = sin (2*x + 3)
f(x)=sin⁡2(2x+3)f{\left(x \right)} = \sin^{2}{\left(2 x + 3 \right)} 
f = sin(2*x + 3)^2
The graph of the function
02468-8-6-4-2-101002
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⁡2(2x+3)=0\sin^{2}{\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=−64.3318529519991x_{1} = -64.3318529519991
x2=20.4911484645524x_{2} = 20.4911484645524
x3=72.3274273108169x_{3} = 72.3274273108169
x4=64.4734456188499x_{4} = 64.4734456188499
x5=−23.4911486889389x_{5} = -23.4911486889389
x6=58.1902604731576x_{6} = 58.1902604731576
x7=−94.1769831742407x_{7} = -94.1769831742407
x8=37.7699082889724x_{8} = 37.7699082889724
x9=−34.4867229570468x_{9} = -34.4867229570468
x10=53.4778711326136x_{10} = 53.4778711326136
x11=−6.21238886463505x_{11} = -6.21238886463505
x12=−31.3451302949883x_{12} = -31.3451302949883
x13=−89.464594417458x_{13} = -89.464594417458
x14=36.1991118880098x_{14} = 36.1991118880098
x15=−42.340704378769x_{15} = -42.340704378769
x16=48.7654827270239x_{16} = 48.7654827270239
x17=73.8982237628936x_{17} = 73.8982237628936
x18=45.6238899091033x_{18} = 45.6238899091033
x19=−53.3362788620348x_{19} = -53.3362788620348
x20=89.6061870489026x_{20} = 89.6061870489026
x21=88.0353905684316x_{21} = 88.0353905684316
x22=−40.7699076747787x_{22} = -40.7699076747787
x23=15.7787597122709x_{23} = 15.7787597122709
x24=−28.2035374417416x_{24} = -28.2035374417416
x25=−1.50000011229917x_{25} = -1.50000011229917
x26=−86.3230015256305x_{26} = -86.3230015256305
x27=95.8893723423514x_{27} = 95.8893723423514
x28=−29.7743338628789x_{28} = -29.7743338628789
x29=−67.4734458415332x_{29} = -67.4734458415332
x30=59.7610568654647x_{30} = 59.7610568654647
x31=70.7566312418352x_{31} = 70.7566312418352
x32=81.752205441742x_{32} = 81.752205441742
x33=−15.6371670074559x_{33} = -15.6371670074559
x34=0.0707961048006144x_{34} = 0.0707961048006144
x35=−59.6194641657212x_{35} = -59.6194641657212
x36=31.4867226097166x_{36} = 31.4867226097166
x37=−95.7477796093333x_{37} = -95.7477796093333
x38=23.6327413380252x_{38} = 23.6327413380252
x39=100.601761129251x_{39} = 100.601761129251
x40=44.0530933625745x_{40} = 44.0530933625745
x41=56.6194639846237x_{41} = 56.6194639846237
x42=−78.4690200383061x_{42} = -78.4690200383061
x43=−58.0486677198505x_{43} = -58.0486677198505
x44=78.6106125566237x_{44} = 78.6106125566237
x45=−72.185834596546x_{45} = -72.185834596546
x46=−7.78318527986359x_{46} = -7.78318527986359
x47=4.78318549937173x_{47} = 4.78318549937173
x48=6.35398157189193x_{48} = 6.35398157189193
x49=−36.0575191385333x_{49} = -36.0575191385333
x50=−51.7654824454163x_{50} = -51.7654824454163
x51=−87.8937975071267x_{51} = -87.8937975071267
x52=−75.3274274453509x_{52} = -75.3274274453509
x53=94.3185758907642x_{53} = 94.3185758907642
x54=86.4645941962783x_{54} = 86.4645941962783
x55=51.9070751831885x_{55} = 51.9070751831885
x56=−73.7566310275489x_{56} = -73.7566310275489
x57=−37.6283155866601x_{57} = -37.6283155866601
x58=−12.4955744275382x_{58} = -12.4955744275382
x59=34.6283154133407x_{59} = 34.6283154133407
x60=75.4690197217621x_{60} = 75.4690197217621
x61=−804.176923076907x_{61} = -804.176923076907
x62=−80.0398163016198x_{62} = -80.0398163016198
x63=22.061944743021x_{63} = 22.061944743021
x64=29.9159266032184x_{64} = 29.9159266032184
x65=−20.349555805984x_{65} = -20.349555805984
x66=−34.4867227269966x_{66} = -34.4867227269966
x67=−14.066370557611x_{67} = -14.066370557611
x68=−45.4822972653549x_{68} = -45.4822972653549
x69=−94.1769832428847x_{69} = -94.1769832428847
x70=92.747779760753x_{70} = 92.747779760753
x71=−9.35398172661415x_{71} = -9.35398172661415
x72=95.8893725783361x_{72} = 95.8893725783361
x73=−100.460168587817x_{73} = -100.460168587817
x74=9.49557409387618x_{74} = 9.49557409387618
x75=12.6371668428837x_{75} = 12.6371668428837
x76=26.7743341947881x_{76} = 26.7743341947881
x77=28.3451301513849x_{77} = 28.3451301513849
x78=−81.610612744639x_{78} = -81.610612744639
x79=67.6150384794371x_{79} = 67.6150384794371
x80=−75.3274274274989x_{80} = -75.3274274274989
x81=14.2079633037244x_{81} = 14.2079633037244
x82=−97.3185759910612x_{82} = -97.3185759910612
x83=42.4822970416052x_{83} = 42.4822970416052
x84=−50.1946860190459x_{84} = -50.1946860190459
x85=−56.4778714943287x_{85} = -56.4778714943287
x86=80.1814090593883x_{86} = 80.1814090593883
x87=50.3362787310254x_{87} = 50.3362787310254
x88=66.0442419694003x_{88} = 66.0442419694003
x89=1.64159276630259x_{89} = 1.64159276630259
x90=7.92477802296118x_{90} = 7.92477802296118
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)^2.
sin⁡2(0⋅2+3)\sin^{2}{\left(0 \cdot 2 + 3 \right)}
The result:
f(0)=sin⁡2(3)f{\left(0 \right)} = \sin^{2}{\left(3 \right)}
The point:
(0, sin(3)^2)
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
4sin⁡(2x+3)cos⁡(2x+3)=04 \sin{\left(2 x + 3 \right)} \cos{\left(2 x + 3 \right)} = 0
Solve this equation
The roots of this equation
x1=−32x_{1} = - \frac{3}{2}
x2=−32−π4x_{2} = - \frac{3}{2} - \frac{\pi}{4}
x3=−32+π4x_{3} = - \frac{3}{2} + \frac{\pi}{4}
The values of the extrema at the points:
(-3/2, 0)

   3   pi    
(- - - --, 1)
   2   4     

   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=−32x_{1} = - \frac{3}{2}
Maxima of the function at points:
x1=−32−π4x_{1} = - \frac{3}{2} - \frac{\pi}{4}
x1=−32+π4x_{1} = - \frac{3}{2} + \frac{\pi}{4}
Decreasing at intervals
(−∞,−32−π4]∪[−32,∞)\left(-\infty, - \frac{3}{2} - \frac{\pi}{4}\right] \cup \left[- \frac{3}{2}, \infty\right)
Increasing at intervals
(−∞,−32]∪[−32+π4,∞)\left(-\infty, - \frac{3}{2}\right] \cup \left[- \frac{3}{2} + \frac{\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
8(−sin⁡2(2x+3)+cos⁡2(2x+3))=08 \left(- \sin^{2}{\left(2 x + 3 \right)} + \cos^{2}{\left(2 x + 3 \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−32−π8x_{1} = - \frac{3}{2} - \frac{\pi}{8}
x2=−32+π8x_{2} = - \frac{3}{2} + \frac{\pi}{8}

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−π8,−32+π8]\left[- \frac{3}{2} - \frac{\pi}{8}, - \frac{3}{2} + \frac{\pi}{8}\right]
Convex at the intervals
(−∞,−32−π8]∪[−32+π8,∞)\left(-\infty, - \frac{3}{2} - \frac{\pi}{8}\right] \cup \left[- \frac{3}{2} + \frac{\pi}{8}, \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→−∞sin⁡2(2x+3)=⟨0,1⟩\lim_{x \to -\infty} \sin^{2}{\left(2 x + 3 \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞sin⁡2(2x+3)=⟨0,1⟩\lim_{x \to \infty} \sin^{2}{\left(2 x + 3 \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x + 3)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(2x+3)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\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⁡2(2x+3)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\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⁡2(2x+3)=sin⁡2(2x−3)\sin^{2}{\left(2 x + 3 \right)} = \sin^{2}{\left(2 x - 3 \right)}
- No
sin⁡2(2x+3)=−sin⁡2(2x−3)\sin^{2}{\left(2 x + 3 \right)} = - \sin^{2}{\left(2 x - 3 \right)}
- No
so, the function
not is
neither even, nor odd
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