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Plotting a function graph/ 4sin^2x

Graphing y = 4sin^2x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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            2   
f(x) = 4*sin (x)
f(x)=4sin⁡2(x)f{\left(x \right)} = 4 \sin^{2}{\left(x \right)} 
f = 4*sin(x)^2
The graph of the function
0.000.250.500.751.001.251.501.752.002.252.502.753.0005
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:
4sin⁡2(x)=04 \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=−47.123890151099x_{1} = -47.123890151099
x2=−37.6991118771514x_{2} = -37.6991118771514
x3=91.1061867314459x_{3} = 91.1061867314459
x4=−9.42477812668337x_{4} = -9.42477812668337
x5=−69.1150386253436x_{5} = -69.1150386253436
x6=−56.5486675191652x_{6} = -56.5486675191652
x7=97.3893725148693x_{7} = 97.3893725148693
x8=−6.28318513794069x_{8} = -6.28318513794069
x9=−91.1061872003049x_{9} = -91.1061872003049
x10=−62.8318528379059x_{10} = -62.8318528379059
x11=−47.1238900492539x_{11} = -47.1238900492539
x12=−34.5575189426108x_{12} = -34.5575189426108
x13=40.8407042560881x_{13} = 40.8407042560881
x14=37.6991120192083x_{14} = 37.6991120192083
x15=31.4159267865366x_{15} = 31.4159267865366
x16=−97.3893724403711x_{16} = -97.3893724403711
x17=50.2654824463473x_{17} = 50.2654824463473
x18=−18.8495556944209x_{18} = -18.8495556944209
x19=−94.2477794529919x_{19} = -94.2477794529919
x20=75.3982241944528x_{20} = 75.3982241944528
x21=69.1150381602162x_{21} = 69.1150381602162
x22=−1734.15914475848x_{22} = -1734.15914475848
x23=97.3893727097471x_{23} = 97.3893727097471
x24=−87.9645943587732x_{24} = -87.9645943587732
x25=91.1061871583643x_{25} = 91.1061871583643
x26=−100.530964672522x_{26} = -100.530964672522
x27=−62.8318532583801x_{27} = -62.8318532583801
x28=−75.3982238620294x_{28} = -75.3982238620294
x29=72.256631027719x_{29} = 72.256631027719
x30=9.42477821024198x_{30} = 9.42477821024198
x31=75.3982239388525x_{31} = 75.3982239388525
x32=0x_{32} = 0
x33=69.1150385885879x_{33} = 69.1150385885879
x34=−31.4159267959754x_{34} = -31.4159267959754
x35=−65.9734457650176x_{35} = -65.9734457650176
x36=−28.2743337166085x_{36} = -28.2743337166085
x37=3.14159287686128x_{37} = 3.14159287686128
x38=−12.5663703661411x_{38} = -12.5663703661411
x39=−72.2566308741333x_{39} = -72.2566308741333
x40=−18.8495561207399x_{40} = -18.8495561207399
x41=62.8318528326557x_{41} = 62.8318528326557
x42=−15.7079632965264x_{42} = -15.7079632965264
x43=94.2477796093525x_{43} = 94.2477796093525
x44=53.4070756765307x_{44} = 53.4070756765307
x45=−84.8230018263493x_{45} = -84.8230018263493
x46=6.28318528425126x_{46} = 6.28318528425126
x47=−106.814150357553x_{47} = -106.814150357553
x48=34.5575190304759x_{48} = 34.5575190304759
x49=−3.14159289677385x_{49} = -3.14159289677385
x50=−25.132741473063x_{50} = -25.132741473063
x51=−78.5398160958028x_{51} = -78.5398160958028
x52=56.5486676091327x_{52} = 56.5486676091327
x53=47.123889589354x_{53} = 47.123889589354
x54=−84.82300141007x_{54} = -84.82300141007
x55=−43.9822971745789x_{55} = -43.9822971745789
x56=25.1327414478072x_{56} = 25.1327414478072
x57=18.8495554002244x_{57} = 18.8495554002244
x58=−53.4070752836338x_{58} = -53.4070752836338
x59=−25.132741632083x_{59} = -25.132741632083
x60=25.1327410188866x_{60} = 25.1327410188866
x61=3.14159244884412x_{61} = 3.14159244884412
x62=−3.14159311568248x_{62} = -3.14159311568248
x63=59.6902605976901x_{63} = 59.6902605976901
x64=−40.8407046898283x_{64} = -40.8407046898283
x65=−50.2654822953391x_{65} = -50.2654822953391
x66=−31.4159267051849x_{66} = -31.4159267051849
x67=15.7079634406648x_{67} = 15.7079634406648
x68=−69.1150386737158x_{68} = -69.1150386737158
x69=47.123890018392x_{69} = 47.123890018392
x70=21.9911485851964x_{70} = 21.9911485851964
x71=53.4070753627408x_{71} = 53.4070753627408
x72=78.5398161878405x_{72} = 78.5398161878405
x73=28.2743338652012x_{73} = 28.2743338652012
x74=81.6814091761104x_{74} = 81.6814091761104
x75=−21.9911485864515x_{75} = -21.9911485864515
x76=−59.6902604576401x_{76} = -59.6902604576401
x77=65.9734457528975x_{77} = 65.9734457528975
x78=−81.6814090380061x_{78} = -81.6814090380061
x79=100.530964766599x_{79} = 100.530964766599
x80=−40.8407042660168x_{80} = -40.8407042660168
x81=87.9645943357576x_{81} = 87.9645943357576
x82=84.8230014093114x_{82} = 84.8230014093114
x83=−91.106187201329x_{83} = -91.106187201329
x84=18.8495556796107x_{84} = 18.8495556796107
x85=12.5663704518704x_{85} = 12.5663704518704
x86=−34.5575189701076x_{86} = -34.5575189701076
x87=43.982297169427x_{87} = 43.982297169427
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4*sin(x)^2.
4sin⁡2(0)4 \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
8sin⁡(x)cos⁡(x)=08 \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     
(----, 4)
  2      

 pi    
(--, 4)
 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
8(−sin⁡2(x)+cos⁡2(x))=08 \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→−∞(4sin⁡2(x))=⟨0,4⟩\lim_{x \to -\infty}\left(4 \sin^{2}{\left(x \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
lim⁡x→∞(4sin⁡2(x))=⟨0,4⟩\lim_{x \to \infty}\left(4 \sin^{2}{\left(x \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4*sin(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(4sin⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{4 \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→∞(4sin⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{4 \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:
4sin⁡2(x)=4sin⁡2(x)4 \sin^{2}{\left(x \right)} = 4 \sin^{2}{\left(x \right)}
- Yes
4sin⁡2(x)=−4sin⁡2(x)4 \sin^{2}{\left(x \right)} = - 4 \sin^{2}{\left(x \right)}
- No
so, the function
is
even
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