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Plotting a function graph/ sin(x)^2

Graphing y = sin(x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2   
f(x) = sin (x)
f(x)=sin⁡2(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} 
f = sin(x)^2
The graph of the function
0.000.250.500.751.001.251.501.752.002.252.502.753.0002
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)=0\sin^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−40.8407046898283x_{1} = -40.8407046898283
x2=−47.123890151099x_{2} = -47.123890151099
x3=72.256631027719x_{3} = 72.256631027719
x4=−78.5398160958028x_{4} = -78.5398160958028
x5=91.1061871583643x_{5} = 91.1061871583643
x6=−1734.15914475848x_{6} = -1734.15914475848
x7=−18.8495561207399x_{7} = -18.8495561207399
x8=−12.5663703661411x_{8} = -12.5663703661411
x9=−59.6902604576401x_{9} = -59.6902604576401
x10=100.530964766599x_{10} = 100.530964766599
x11=9.42477859080277x_{11} = 9.42477859080277
x12=−34.5575189426108x_{12} = -34.5575189426108
x13=−47.1238900492539x_{13} = -47.1238900492539
x14=12.5663704518704x_{14} = 12.5663704518704
x15=−106.814150357553x_{15} = -106.814150357553
x16=−12.5663700417108x_{16} = -12.5663700417108
x17=−97.3893724403711x_{17} = -97.3893724403711
x18=43.982297169427x_{18} = 43.982297169427
x19=−43.9822971745789x_{19} = -43.9822971745789
x20=18.8495554002244x_{20} = 18.8495554002244
x21=−69.1150386737158x_{21} = -69.1150386737158
x22=−84.82300141007x_{22} = -84.82300141007
x23=25.1327410188866x_{23} = 25.1327410188866
x24=−65.9734457650176x_{24} = -65.9734457650176
x25=15.7079634406648x_{25} = 15.7079634406648
x26=−53.4070752836338x_{26} = -53.4070752836338
x27=81.6814091761104x_{27} = 81.6814091761104
x28=84.8230014093114x_{28} = 84.8230014093114
x29=62.8318528326557x_{29} = 62.8318528326557
x30=56.5486676091327x_{30} = 56.5486676091327
x31=−34.5575189701076x_{31} = -34.5575189701076
x32=75.3982239388525x_{32} = 75.3982239388525
x33=9.42477821024198x_{33} = 9.42477821024198
x34=50.2654824463473x_{34} = 50.2654824463473
x35=−87.9645943587732x_{35} = -87.9645943587732
x36=−6.28318513794069x_{36} = -6.28318513794069
x37=53.4070756765307x_{37} = 53.4070756765307
x38=−21.9911485864515x_{38} = -21.9911485864515
x39=37.6991120192083x_{39} = 37.6991120192083
x40=62.8318524523063x_{40} = 62.8318524523063
x41=40.840703919946x_{41} = 40.840703919946
x42=3.14159287686128x_{42} = 3.14159287686128
x43=18.8495556796107x_{43} = 18.8495556796107
x44=78.5398161878405x_{44} = 78.5398161878405
x45=−94.2477794529919x_{45} = -94.2477794529919
x46=−81.6814090380061x_{46} = -81.6814090380061
x47=−62.8318528379059x_{47} = -62.8318528379059
x48=−69.1150386253436x_{48} = -69.1150386253436
x49=−37.6991118771514x_{49} = -37.6991118771514
x50=25.1327414478072x_{50} = 25.1327414478072
x51=−25.132741632083x_{51} = -25.132741632083
x52=6.28318528425126x_{52} = 6.28318528425126
x53=−100.530964672522x_{53} = -100.530964672522
x54=87.9645943357576x_{54} = 87.9645943357576
x55=97.3893725148693x_{55} = 97.3893725148693
x56=−40.8407042660168x_{56} = -40.8407042660168
x57=−31.4159267051849x_{57} = -31.4159267051849
x58=−31.4159267959754x_{58} = -31.4159267959754
x59=47.123889589354x_{59} = 47.123889589354
x60=0x_{60} = 0
x61=−84.8230018263493x_{61} = -84.8230018263493
x62=−15.7079632965264x_{62} = -15.7079632965264
x63=−62.8318532583801x_{63} = -62.8318532583801
x64=−3.14159289677385x_{64} = -3.14159289677385
x65=−72.2566308741333x_{65} = -72.2566308741333
x66=91.1061867314459x_{66} = 91.1061867314459
x67=69.1150385885879x_{67} = 69.1150385885879
x68=−9.42477812668337x_{68} = -9.42477812668337
x69=47.123890018392x_{69} = 47.123890018392
x70=75.3982241944528x_{70} = 75.3982241944528
x71=3.14159244884412x_{71} = 3.14159244884412
x72=40.8407042560881x_{72} = 40.8407042560881
x73=−25.132741473063x_{73} = -25.132741473063
x74=−91.106187201329x_{74} = -91.106187201329
x75=−50.2654822953391x_{75} = -50.2654822953391
x76=94.2477796093525x_{76} = 94.2477796093525
x77=34.5575190304759x_{77} = 34.5575190304759
x78=21.9911485851964x_{78} = 21.9911485851964
x79=65.9734457528975x_{79} = 65.9734457528975
x80=−91.1061872003049x_{80} = -91.1061872003049
x81=69.1150381602162x_{81} = 69.1150381602162
x82=−56.5486675191652x_{82} = -56.5486675191652
x83=84.8230010166547x_{83} = 84.8230010166547
x84=−18.8495556944209x_{84} = -18.8495556944209
x85=−3.14159311568248x_{85} = -3.14159311568248
x86=31.4159267865366x_{86} = 31.4159267865366
x87=59.6902605976901x_{87} = 59.6902605976901
x88=97.3893727097471x_{88} = 97.3893727097471
x89=53.4070753627408x_{89} = 53.4070753627408
x90=28.2743338652012x_{90} = 28.2743338652012
x91=−28.2743337166085x_{91} = -28.2743337166085
x92=31.4159271479423x_{92} = 31.4159271479423
x93=−75.3982238620294x_{93} = -75.3982238620294
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⁡2(0)\sin^{2}{\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)=02 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 0)

 -pi     
(----, 1)
  2      

 pi    
(--, 1)
 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=0x_{1} = 0
Maxima of the function at points:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,−π2]∪[0,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \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
2(−sin⁡2(x)+cos⁡2(x))=02 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}

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
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Convex at the intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \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(x)=⟨0,1⟩\lim_{x \to -\infty} \sin^{2}{\left(x \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(x)=⟨0,1⟩\lim_{x \to \infty} \sin^{2}{\left(x \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(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\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)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\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⁡2(x)\sin^{2}{\left(x \right)} = \sin^{2}{\left(x \right)}
- Yes
sin⁡2(x)=−sin⁡2(x)\sin^{2}{\left(x \right)} = - \sin^{2}{\left(x \right)}
- No
so, the function
is
even
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