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Plotting a function graph/ (x^3)/(4-x^2)

Graphing y = (x^3)/(4-x^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          3  
         x   
f(x) = ------
            2
       4 - x
f(x)=x34−x2f{\left(x \right)} = \frac{x^{3}}{4 - x^{2}} 
f = x^3/(4 - x^2)
The graph of the function
02468-8-6-4-2-1010-100100
The domain of the function
The points at which the function is not precisely defined:
x1=−2x_{1} = -2
x2=2x_{2} = 2
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:
x34−x2=0\frac{x^{3}}{4 - x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−9.55409138170962⋅10−5x_{1} = -9.55409138170962 \cdot 10^{-5}
x2=4.24641013841956⋅10−5x_{2} = 4.24641013841956 \cdot 10^{-5}
x3=−3.98561068030404⋅10−5x_{3} = -3.98561068030404 \cdot 10^{-5}
x4=3.49392143580647⋅10−5x_{4} = 3.49392143580647 \cdot 10^{-5}
x5=−3.39728377317918⋅10−5x_{5} = -3.39728377317918 \cdot 10^{-5}
x6=3.4077552501763⋅10−5x_{6} = 3.4077552501763 \cdot 10^{-5}
x7=−3.76797425836015⋅10−5x_{7} = -3.76797425836015 \cdot 10^{-5}
x8=8.99412147700574⋅10−5x_{8} = 8.99412147700574 \cdot 10^{-5}
x9=7.49788349400003⋅10−5x_{9} = 7.49788349400003 \cdot 10^{-5}
x10=5.63958508047232⋅10−5x_{10} = 5.63958508047232 \cdot 10^{-5}
x11=3.32575527575275⋅10−5x_{11} = 3.32575527575275 \cdot 10^{-5}
x12=−5.39025886053056⋅10−5x_{12} = -5.39025886053056 \cdot 10^{-5}
x13=4.67917831062956⋅10−5x_{13} = 4.67917831062956 \cdot 10^{-5}
x14=−2.89880807553215⋅10−5x_{14} = -2.89880807553215 \cdot 10^{-5}
x15=3.78087380223353⋅10−5x_{15} = 3.78087380223353 \cdot 10^{-5}
x16=4.38141824340841⋅10−5x_{16} = 4.38141824340841 \cdot 10^{-5}
x17=−0.000170166696018749x_{17} = -0.000170166696018749
x18=−3.57299101650919⋅10−5x_{18} = -3.57299101650919 \cdot 10^{-5}
x19=6.43518491947289⋅10−5x_{19} = 6.43518491947289 \cdot 10^{-5}
x20=3.03388703893791⋅10−5x_{20} = 3.03388703893791 \cdot 10^{-5}
x21=−3.87371963276928⋅10−5x_{21} = -3.87371963276928 \cdot 10^{-5}
x22=−3.31578314658875⋅10−5x_{22} = -3.31578314658875 \cdot 10^{-5}
x23=−3.09325284157027⋅10−5x_{23} = -3.09325284157027 \cdot 10^{-5}
x24=3.58458128063165⋅10−5x_{24} = 3.58458128063165 \cdot 10^{-5}
x25=−3.02559240383893⋅10−5x_{25} = -3.02559240383893 \cdot 10^{-5}
x26=−7.05989124402088⋅10−5x_{26} = -7.05989124402088 \cdot 10^{-5}
x27=−5.85027031873737⋅10−5x_{27} = -5.85027031873737 \cdot 10^{-5}
x28=3.68009765524614⋅10−5x_{28} = 3.68009765524614 \cdot 10^{-5}
x29=−7.87896531803127⋅10−5x_{29} = -7.87896531803127 \cdot 10^{-5}
x30=−4.50685286545851⋅10−5x_{30} = -4.50685286545851 \cdot 10^{-5}
x31=−3.48291195273371⋅10−5x_{31} = -3.48291195273371 \cdot 10^{-5}
x32=−0.000102895975701088x_{32} = -0.000102895975701088
x33=−0.000150015789823077x_{33} = -0.000150015789823077
x34=4.52536561302307⋅10−5x_{34} = 4.52536561302307 \cdot 10^{-5}
x35=0.000103898950746591x_{35} = 0.000103898950746591
x36=−2.7822538545061⋅10−5x_{36} = -2.7822538545061 \cdot 10^{-5}
x37=4.11952833663906⋅10−5x_{37} = 4.11952833663906 \cdot 10^{-5}
x38=3.24762525114847⋅10−5x_{38} = 3.24762525114847 \cdot 10^{-5}
x39=−3.66787915250259⋅10−5x_{39} = -3.66787915250259 \cdot 10^{-5}
x40=−4.36407104791292⋅10−5x_{40} = -4.36407104791292 \cdot 10^{-5}
x41=−2.83933143595082⋅10−5x_{41} = -2.83933143595082 \cdot 10^{-5}
x42=−0.000111533956499186x_{42} = -0.000111533956499186
x43=8.43148385999583⋅10−5x_{43} = 8.43148385999583 \cdot 10^{-5}
x44=3.8873591438861⋅10−5x_{44} = 3.8873591438861 \cdot 10^{-5}
x45=4.00005612806713⋅10−5x_{45} = 4.00005612806713 \cdot 10^{-5}
x46=6.14581028280139⋅10−5x_{46} = 6.14581028280139 \cdot 10^{-5}
x47=−5.61073488359365⋅10−5x_{47} = -5.61073488359365 \cdot 10^{-5}
x48=−4.65937740593902⋅10−5x_{48} = -4.65937740593902 \cdot 10^{-5}
x49=−6.39750784194713⋅10−5x_{49} = -6.39750784194713 \cdot 10^{-5}
x50=−4.99797702619475⋅10−5x_{50} = -4.99797702619475 \cdot 10^{-5}
x51=5.2112311637843⋅10−5x_{51} = 5.2112311637843 \cdot 10^{-5}
x52=7.10596406008811⋅10−5x_{52} = 7.10596406008811 \cdot 10^{-5}
x53=0.000136141239154743x_{53} = 0.000136141239154743
x54=−3.23811744285416⋅10−5x_{54} = -3.23811744285416 \cdot 10^{-5}
x55=−6.71211256073378⋅10−5x_{55} = -6.71211256073378 \cdot 10^{-5}
x56=0x_{56} = 0
x57=9.64002910137799⋅10−5x_{57} = 9.64002910137799 \cdot 10^{-5}
x58=4.84391604953844⋅10−5x_{58} = 4.84391604953844 \cdot 10^{-5}
x59=−0.000121839971471264x_{59} = -0.000121839971471264
x60=6.75366612766008⋅10−5x_{60} = 6.75366612766008 \cdot 10^{-5}
x61=0.000123271008137431x_{61} = 0.000123271008137431
x62=−3.16402116018595⋅10−5x_{62} = -3.16402116018595 \cdot 10^{-5}
x63=−8.91959720651235⋅10−5x_{63} = -8.91959720651235 \cdot 10^{-5}
x64=2.78926234301106⋅10−5x_{64} = 2.78926234301106 \cdot 10^{-5}
x65=3.17309646140084⋅10−5x_{65} = 3.17309646140084 \cdot 10^{-5}
x66=−5.18663227219019⋅10−5x_{66} = -5.18663227219019 \cdot 10^{-5}
x67=2.96878048887459⋅10−5x_{67} = 2.96878048887459 \cdot 10^{-5}
x68=−4.23012106715333⋅10−5x_{68} = -4.23012106715333 \cdot 10^{-5}
x69=−8.36619469563777⋅10−5x_{69} = -8.36619469563777 \cdot 10^{-5}
x70=−7.4464953356113⋅10−5x_{70} = -7.4464953356113 \cdot 10^{-5}
x71=−0.000134378358346033x_{71} = -0.000134378358346033
x72=−2.96083881614977⋅10−5x_{72} = -2.96083881614977 \cdot 10^{-5}
x73=−4.10420284793804⋅10−5x_{73} = -4.10420284793804 \cdot 10^{-5}
x74=7.93666780143691⋅10−5x_{74} = 7.93666780143691 \cdot 10^{-5}
x75=−6.11148468260592⋅10−5x_{75} = -6.11148468260592 \cdot 10^{-5}
x76=0.00011272150063151x_{76} = 0.00011272150063151
x77=5.41685505962669⋅10−5x_{77} = 5.41685505962669 \cdot 10^{-5}
x78=0.000152250219301269x_{78} = 0.000152250219301269
x79=2.8466317616625⋅10−5x_{79} = 2.8466317616625 \cdot 10^{-5}
x80=3.10192459777013⋅10−5x_{80} = 3.10192459777013 \cdot 10^{-5}
x81=−4.82268640354376⋅10−5x_{81} = -4.82268640354376 \cdot 10^{-5}
x82=5.88167774091302⋅10−5x_{82} = 5.88167774091302 \cdot 10^{-5}
x83=5.02079742868411⋅10−5x_{83} = 5.02079742868411 \cdot 10^{-5}
x84=2.90641891779644⋅10−5x_{84} = 2.90641891779644 \cdot 10^{-5}
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^3/(4 - x^2).
034−02\frac{0^{3}}{4 - 0^{2}}
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
2x4(4−x2)2+3x24−x2=0\frac{2 x^{4}}{\left(4 - x^{2}\right)^{2}} + \frac{3 x^{2}}{4 - x^{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−23x_{2} = - 2 \sqrt{3}
x3=23x_{3} = 2 \sqrt{3}
The values of the extrema at the points:
(0, 0)

      ___      ___ 
(-2*\/ 3, 3*\/ 3 )

     ___       ___ 
(2*\/ 3, -3*\/ 3 )


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=−23x_{1} = - 2 \sqrt{3}
Maxima of the function at points:
x1=23x_{1} = 2 \sqrt{3}
Decreasing at intervals
[−23,23]\left[- 2 \sqrt{3}, 2 \sqrt{3}\right]
Increasing at intervals
(−∞,−23]∪[23,∞)\left(-\infty, - 2 \sqrt{3}\right] \cup \left[2 \sqrt{3}, \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
2x(−x2(4x2x2−4−1)x2−4+6x2x2−4−3)x2−4=0\frac{2 x \left(- \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} + \frac{6 x^{2}}{x^{2} - 4} - 3\right)}{x^{2} - 4} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=−2x_{1} = -2
x2=2x_{2} = 2

lim⁡x→−2−(2x(−x2(4x2x2−4−1)x2−4+6x2x2−4−3)x2−4)=∞\lim_{x \to -2^-}\left(\frac{2 x \left(- \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} + \frac{6 x^{2}}{x^{2} - 4} - 3\right)}{x^{2} - 4}\right) = \infty
lim⁡x→−2+(2x(−x2(4x2x2−4−1)x2−4+6x2x2−4−3)x2−4)=−∞\lim_{x \to -2^+}\left(\frac{2 x \left(- \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} + \frac{6 x^{2}}{x^{2} - 4} - 3\right)}{x^{2} - 4}\right) = -\infty
- the limits are not equal, so
x1=−2x_{1} = -2
- is an inflection point
lim⁡x→2−(2x(−x2(4x2x2−4−1)x2−4+6x2x2−4−3)x2−4)=∞\lim_{x \to 2^-}\left(\frac{2 x \left(- \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} + \frac{6 x^{2}}{x^{2} - 4} - 3\right)}{x^{2} - 4}\right) = \infty
lim⁡x→2+(2x(−x2(4x2x2−4−1)x2−4+6x2x2−4−3)x2−4)=−∞\lim_{x \to 2^+}\left(\frac{2 x \left(- \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right)}{x^{2} - 4} + \frac{6 x^{2}}{x^{2} - 4} - 3\right)}{x^{2} - 4}\right) = -\infty
- the limits are not equal, so
x2=2x_{2} = 2
- is an inflection point

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Vertical asymptotes
Have:
x1=−2x_{1} = -2
x2=2x_{2} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x34−x2)=∞\lim_{x \to -\infty}\left(\frac{x^{3}}{4 - x^{2}}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x34−x2)=−∞\lim_{x \to \infty}\left(\frac{x^{3}}{4 - x^{2}}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^3/(4 - x^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x24−x2)=−1\lim_{x \to -\infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the left:
y=−xy = - x
lim⁡x→∞(x24−x2)=−1\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the right:
y=−xy = - x
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:
x34−x2=−x34−x2\frac{x^{3}}{4 - x^{2}} = - \frac{x^{3}}{4 - x^{2}}
- No
x34−x2=x34−x2\frac{x^{3}}{4 - x^{2}} = \frac{x^{3}}{4 - x^{2}}
- Yes
so, the function
is
odd
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