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Plotting a function graph/ sin^2x+sinx

Graphing y = sin^2x+sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin (x) + sin(x)
f(x)=sin⁡2(x)+sin⁡(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} + \sin{\left(x \right)} 
f = sin(x)^2 + sin(x)
The graph of the function
02468-8-6-4-2-10102.5-2.5
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:
sin⁡2(x)+sin⁡(x)=0\sin^{2}{\left(x \right)} + \sin{\left(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}
x3=πx_{3} = \pi
x4=3π2x_{4} = \frac{3 \pi}{2}
Numerical solution
x1=−18.8495559215388x_{1} = -18.8495559215388
x2=−89.5353907430632x_{2} = -89.5353907430632
x3=29.8451303173708x_{3} = 29.8451303173708
x4=−15.707963267949x_{4} = -15.707963267949
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=−45.5530935853668x_{6} = -45.5530935853668
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=84.8230016469244x_{8} = 84.8230016469244
x9=10.9955738002317x_{9} = 10.9955738002317
x10=21.9911485751286x_{10} = 21.9911485751286
x11=0x_{11} = 0
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=98.9601685293103x_{13} = 98.9601685293103
x14=28.2743338823081x_{14} = 28.2743338823081
x15=98.9601686349023x_{15} = 98.9601686349023
x16=4.71238880126794x_{16} = 4.71238880126794
x17=72.2566310325652x_{17} = 72.2566310325652
x18=80.1106131505346x_{18} = 80.1106131505346
x19=−32.9867232359983x_{19} = -32.9867232359983
x20=−89.5353901159531x_{20} = -89.5353901159531
x21=−70.6858346750384x_{21} = -70.6858346750384
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=86.3937978896245x_{24} = 86.3937978896245
x25=87.9645943005142x_{25} = 87.9645943005142
x26=−7.85398150061792x_{26} = -7.85398150061792
x27=−50.2654824574367x_{27} = -50.2654824574367
x28=−43.9822971502571x_{28} = -43.9822971502571
x29=−97.3893722612836x_{29} = -97.3893722612836
x30=50.2654824574367x_{30} = 50.2654824574367
x31=59.6902604182061x_{31} = 59.6902604182061
x32=−91.106186954104x_{32} = -91.106186954104
x33=−76.9690201700248x_{33} = -76.9690201700248
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=−51.8362785033387x_{35} = -51.8362785033387
x36=12.5663706143592x_{36} = 12.5663706143592
x37=−81.6814089933346x_{37} = -81.6814089933346
x38=54.9778715673102x_{38} = 54.9778715673102
x39=−83.2522054990561x_{39} = -83.2522054990561
x40=94.2477796076938x_{40} = 94.2477796076938
x41=81.6814089933346x_{41} = 81.6814089933346
x42=−95.8185758682081x_{42} = -95.8185758682081
x43=73.8274274758159x_{43} = 73.8274274758159
x44=67.5442422387326x_{44} = 67.5442422387326
x45=−1.57079642735461x_{45} = -1.57079642735461
x46=92.6769831047852x_{46} = 92.6769831047852
x47=31.4159265358979x_{47} = 31.4159265358979
x48=−26.7035379972524x_{48} = -26.7035379972524
x49=−28.2743338823081x_{49} = -28.2743338823081
x50=17.2787594994858x_{50} = 17.2787594994858
x51=69.1150383789755x_{51} = 69.1150383789755
x52=−72.2566310325652x_{52} = -72.2566310325652
x53=37.6991118430775x_{53} = 37.6991118430775
x54=65.9734457253857x_{54} = 65.9734457253857
x55=−84.8230016469244x_{55} = -84.8230016469244
x56=−20.4203520633206x_{56} = -20.4203520633206
x57=−26.7035375777291x_{57} = -26.7035375777291
x58=3.14159265358979x_{58} = 3.14159265358979
x59=−87.9645943005142x_{59} = -87.9645943005142
x60=40.8407044966673x_{60} = 40.8407044966673
x61=47.1238898038469x_{61} = 47.1238898038469
x62=91.106186954104x_{62} = 91.106186954104
x63=78.5398163397448x_{63} = 78.5398163397448
x64=−6.28318530717959x_{64} = -6.28318530717959
x65=−64.4026492153213x_{65} = -64.4026492153213
x66=15.707963267949x_{66} = 15.707963267949
x67=−51.8362786901812x_{67} = -51.8362786901812
x68=25.1327412287183x_{68} = 25.1327412287183
x69=−3.14159265358979x_{69} = -3.14159265358979
x70=56.5486677646163x_{70} = 56.5486677646163
x71=−39.2699083493443x_{71} = -39.2699083493443
x72=−100.530964914873x_{72} = -100.530964914873
x73=−58.1194640010631x_{73} = -58.1194640010631
x74=42.4115007309741x_{74} = 42.4115007309741
x75=−76.969019611197x_{75} = -76.969019611197
x76=−62.8318530717959x_{76} = -62.8318530717959
x77=−14.1371668415708x_{77} = -14.1371668415708
x78=36.1283157927187x_{78} = 36.1283157927187
x79=−65.9734457253857x_{79} = -65.9734457253857
x80=−34.5575191894877x_{80} = -34.5575191894877
x81=43.9822971502571x_{81} = 43.9822971502571
x82=−37.6991118430775x_{82} = -37.6991118430775
x83=−59.6902604182061x_{83} = -59.6902604182061
x84=−75.398223686155x_{84} = -75.398223686155
x85=100.530964914873x_{85} = 100.530964914873
x86=10.9955745314484x_{86} = 10.9955745314484
x87=75.398223686155x_{87} = 75.398223686155
x88=34.5575191894877x_{88} = 34.5575191894877
x89=6.28318530717959x_{89} = 6.28318530717959
x90=61.2610565484827x_{90} = 61.2610565484827
x91=23.5619450880496x_{91} = 23.5619450880496
x92=−47.1238898038469x_{92} = -47.1238898038469
x93=−32.9867224730462x_{93} = -32.9867224730462
x94=−9.42477796076938x_{94} = -9.42477796076938
x95=54.9778709836x_{95} = 54.9778709836
x96=61.261057240759x_{96} = 61.261057240759
x97=−12.5663706143592x_{97} = -12.5663706143592
x98=17.2787599718548x_{98} = 17.2787599718548
x99=48.6946859526732x_{99} = 48.6946859526732
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^2 + sin(x).
sin⁡2(0)+sin⁡(0)\sin^{2}{\left(0 \right)} + \sin{\left(0 \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
2sin⁡(x)cos⁡(x)+cos⁡(x)=02 \sin{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−5π6x_{1} = - \frac{5 \pi}{6}
x2=−π2x_{2} = - \frac{\pi}{2}
x3=−π6x_{3} = - \frac{\pi}{6}
x4=π2x_{4} = \frac{\pi}{2}
The values of the extrema at the points:
 -5*pi       
(-----, -1/4)
   6         

 -pi     
(----, 0)
  2      

 -pi        
(----, -1/4)
  6         

 pi    
(--, 2)
 2


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=−5π6x_{1} = - \frac{5 \pi}{6}
x2=−π6x_{2} = - \frac{\pi}{6}
Maxima of the function at points:
x2=−π2x_{2} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Decreasing at intervals
[−5π6,−π2]∪[−π6,∞)\left[- \frac{5 \pi}{6}, - \frac{\pi}{2}\right] \cup \left[- \frac{\pi}{6}, \infty\right)
Increasing at intervals
(−∞,−5π6]\left(-\infty, - \frac{5 \pi}{6}\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
−2sin⁡2(x)−sin⁡(x)+2cos⁡2(x)=0- 2 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−2atan⁡(−14+29−334+334)x_{1} = - 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}
x2=2atan⁡(14+233+94+334)x_{2} = 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{\sqrt{33}}{4} \right)}
x3=2atan⁡(−334+14+29−334)x_{3} = 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}
x4=2atan⁡(−233+94+14+334)x_{4} = 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}

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
[2atan⁡(14+233+94+334),∞)\left[2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{\sqrt{33}}{4} \right)}, \infty\right)
Convex at the intervals
(−∞,2atan⁡(−334+14+29−334)]∪[2atan⁡(−233+94+14+334),2atan⁡(14+233+94+334)]\left(-\infty, 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}\right] \cup \left[2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}, 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{\sqrt{33}}{4} \right)}\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(x)+sin⁡(x))=⟨−1,2⟩\lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
lim⁡x→∞(sin⁡2(x)+sin⁡(x))=⟨−1,2⟩\lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^2 + sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)+sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} + \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡2(x)+sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} + \sin{\left(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:
sin⁡2(x)+sin⁡(x)=sin⁡2(x)−sin⁡(x)\sin^{2}{\left(x \right)} + \sin{\left(x \right)} = \sin^{2}{\left(x \right)} - \sin{\left(x \right)}
- No
sin⁡2(x)+sin⁡(x)=−sin⁡2(x)+sin⁡(x)\sin^{2}{\left(x \right)} + \sin{\left(x \right)} = - \sin^{2}{\left(x \right)} + \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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