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Plotting a function graph/ sin^2x

Graphing y = sin^2x

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