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  • Graphing y =:
  • x^3-((x^4)/4)
  • -x^3+3x+2
  • x^3-3*x-2
  • -x^3-3x^2+3

  • Identical expressions

  • four *sin(x)^ two
  • 4 multiply by sinus of (x) squared
  • four multiply by sinus of (x) to the power of two
  • 4*sin(x)2
  • 4*sinx2
  • 4*sin(x)²
  • 4*sin(x) to the power of 2
  • 4sin(x)^2
  • 4sin(x)2
  • 4sinx2
  • 4sinx^2

  • Similar expressions

  • 4*sinx^2
Plotting a function graph/ 4*sin(x)^2

Graphing y = 4*sin(x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            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
02468-8-6-4-2-101005
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=12.5663704518704x_{1} = 12.5663704518704
x2=−25.132741473063x_{2} = -25.132741473063
x3=−37.6991118771514x_{3} = -37.6991118771514
x4=87.9645943357576x_{4} = 87.9645943357576
x5=−31.4159267051849x_{5} = -31.4159267051849
x6=43.982297169427x_{6} = 43.982297169427
x7=−78.5398160958028x_{7} = -78.5398160958028
x8=6.28318528425126x_{8} = 6.28318528425126
x9=−31.4159267959754x_{9} = -31.4159267959754
x10=−87.9645943587732x_{10} = -87.9645943587732
x11=81.6814091761104x_{11} = 81.6814091761104
x12=−9.42477812668337x_{12} = -9.42477812668337
x13=34.5575190304759x_{13} = 34.5575190304759
x14=100.530964766599x_{14} = 100.530964766599
x15=−47.1238900492539x_{15} = -47.1238900492539
x16=50.2654824463473x_{16} = 50.2654824463473
x17=15.7079634406648x_{17} = 15.7079634406648
x18=−75.3982238620294x_{18} = -75.3982238620294
x19=−15.7079632965264x_{19} = -15.7079632965264
x20=0x_{20} = 0
x21=−65.9734457650176x_{21} = -65.9734457650176
x22=3.14159287686128x_{22} = 3.14159287686128
x23=53.4070753627408x_{23} = 53.4070753627408
x24=−69.1150386253436x_{24} = -69.1150386253436
x25=−106.814150357553x_{25} = -106.814150357553
x26=21.9911485851964x_{26} = 21.9911485851964
x27=−21.9911485864515x_{27} = -21.9911485864515
x28=−3.14159289677385x_{28} = -3.14159289677385
x29=91.1061871583643x_{29} = 91.1061871583643
x30=−72.2566308741333x_{30} = -72.2566308741333
x31=37.6991120192083x_{31} = 37.6991120192083
x32=97.3893727097471x_{32} = 97.3893727097471
x33=−34.5575189426108x_{33} = -34.5575189426108
x34=−53.4070752836338x_{34} = -53.4070752836338
x35=25.1327414478072x_{35} = 25.1327414478072
x36=75.3982241944528x_{36} = 75.3982241944528
x37=−62.8318528379059x_{37} = -62.8318528379059
x38=9.42477821024198x_{38} = 9.42477821024198
x39=84.8230014093114x_{39} = 84.8230014093114
x40=25.1327410188866x_{40} = 25.1327410188866
x41=−43.9822971745789x_{41} = -43.9822971745789
x42=3.14159244884412x_{42} = 3.14159244884412
x43=47.123889589354x_{43} = 47.123889589354
x44=94.2477796093525x_{44} = 94.2477796093525
x45=−62.8318532583801x_{45} = -62.8318532583801
x46=−28.2743337166085x_{46} = -28.2743337166085
x47=−94.2477794529919x_{47} = -94.2477794529919
x48=−18.8495561207399x_{48} = -18.8495561207399
x49=69.1150381602162x_{49} = 69.1150381602162
x50=−1734.15914475848x_{50} = -1734.15914475848
x51=−6.28318513794069x_{51} = -6.28318513794069
x52=−18.8495556944209x_{52} = -18.8495556944209
x53=75.3982239388525x_{53} = 75.3982239388525
x54=−81.6814090380061x_{54} = -81.6814090380061
x55=18.8495554002244x_{55} = 18.8495554002244
x56=69.1150385885879x_{56} = 69.1150385885879
x57=−56.5486675191652x_{57} = -56.5486675191652
x58=−3.14159311568248x_{58} = -3.14159311568248
x59=56.5486676091327x_{59} = 56.5486676091327
x60=−97.3893724403711x_{60} = -97.3893724403711
x61=59.6902605976901x_{61} = 59.6902605976901
x62=78.5398161878405x_{62} = 78.5398161878405
x63=−84.82300141007x_{63} = -84.82300141007
x64=−91.1061872003049x_{64} = -91.1061872003049
x65=31.4159267865366x_{65} = 31.4159267865366
x66=62.8318528326557x_{66} = 62.8318528326557
x67=−91.106187201329x_{67} = -91.106187201329
x68=−34.5575189701076x_{68} = -34.5575189701076
x69=28.2743338652012x_{69} = 28.2743338652012
x70=65.9734457528975x_{70} = 65.9734457528975
x71=−40.8407046898283x_{71} = -40.8407046898283
x72=18.8495556796107x_{72} = 18.8495556796107
x73=91.1061867314459x_{73} = 91.1061867314459
x74=−69.1150386737158x_{74} = -69.1150386737158
x75=−40.8407042660168x_{75} = -40.8407042660168
x76=97.3893725148693x_{76} = 97.3893725148693
x77=40.8407042560881x_{77} = 40.8407042560881
x78=−100.530964672522x_{78} = -100.530964672522
x79=53.4070756765307x_{79} = 53.4070756765307
x80=−50.2654822953391x_{80} = -50.2654822953391
x81=−12.5663703661411x_{81} = -12.5663703661411
x82=47.123890018392x_{82} = 47.123890018392
x83=72.256631027719x_{83} = 72.256631027719
x84=−84.8230018263493x_{84} = -84.8230018263493
x85=−59.6902604576401x_{85} = -59.6902604576401
x86=−25.132741632083x_{86} = -25.132741632083
x87=−47.123890151099x_{87} = -47.123890151099
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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