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Plotting a function graph/ ((x-2)^2)/(x+1)

Graphing y = ((x-2)^2)/(x+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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              2
       (x - 2) 
f(x) = --------
        x + 1
f(x)=(x−2)2x+1f{\left(x \right)} = \frac{\left(x - 2\right)^{2}}{x + 1} 
f = (x - 2)^2/(x + 1)
The graph of the function
02468-8-6-4-2-1010-500500
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -1
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:
(x−2)2x+1=0\frac{\left(x - 2\right)^{2}}{x + 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2x_{1} = 2
Numerical solution
x1=2.00000141987805x_{1} = 2.00000141987805
x2=2.00000100675352x_{2} = 2.00000100675352
x3=2.00000132752919x_{3} = 2.00000132752919
x4=2.00000133379521x_{4} = 2.00000133379521
x5=2.00000142832132x_{5} = 2.00000142832132
x6=2.00000147836967x_{6} = 2.00000147836967
x7=2.00000057455928x_{7} = 2.00000057455928
x8=2.00000142708756x_{8} = 2.00000142708756
x9=2.0000012964472x_{9} = 2.0000012964472
x10=2.00000144742782x_{10} = 2.00000144742782
x11=2.00000145539967x_{11} = 2.00000145539967
x12=2.0000012596859x_{12} = 2.0000012596859
x13=2.00000132224016x_{13} = 2.00000132224016
x14=2.00000128959571x_{14} = 2.00000128959571
x15=2.00000130965461x_{15} = 2.00000130965461
x16=2.00000131747033x_{16} = 2.00000131747033
x17=2.00000130230488x_{17} = 2.00000130230488
x18=2.0000014736478x_{18} = 2.0000014736478
x19=2.00000123549736x_{19} = 2.00000123549736
x20=2.00000127169295x_{20} = 2.00000127169295
x21=2.00000144093121x_{21} = 2.00000144093121
x22=2.00000129948624x_{22} = 2.00000129948624
x23=2.00000131569098x_{23} = 2.00000131569098
x24=2.00000113635187x_{24} = 2.00000113635187
x25=2.00000141901484x_{25} = 2.00000141901484
x26=2.00000132366525x_{26} = 2.00000132366525
x27=2.00000132869617x_{27} = 2.00000132869617
x28=2.00000162249387x_{28} = 2.00000162249387
x29=2.00000116150428x_{29} = 2.00000116150428
x30=2.00000150300774x_{30} = 2.00000150300774
x31=2.00000142479051x_{31} = 2.00000142479051
x32=2.00000130492625x_{32} = 2.00000130492625
x33=2.00000157542508x_{33} = 2.00000157542508
x34=2.00000128147405x_{34} = 2.00000128147405
x35=2.00000092577245x_{35} = 2.00000092577245
x36=2.00000142077767x_{36} = 2.00000142077767
x37=2x_{37} = 2
x38=2.00000119877185x_{38} = 2.00000119877185
x39=2.00000148357777x_{39} = 2.00000148357777
x40=2.00000125259586x_{40} = 2.00000125259586
x41=2.00000184437219x_{41} = 2.00000184437219
x42=2.0000014693471x_{42} = 2.0000014693471
x43=2.00000142371955x_{43} = 2.00000142371955
x44=2.00000143098151x_{44} = 2.00000143098151
x45=2.00000142269575x_{45} = 2.00000142269575
x46=2.00000133087305x_{46} = 2.00000133087305
x47=2.00000110455881x_{47} = 2.00000110455881
x48=2.00000165554121x_{48} = 2.00000165554121
x49=2.00000132980959x_{49} = 2.00000132980959
x50=2.00000143722925x_{50} = 2.00000143722925
x51=2.00000175820848x_{51} = 2.00000175820848
x52=2.00000131380239x_{52} = 2.00000131380239
x53=2.00000146541364x_{53} = 2.00000146541364
x54=2.00000154351297x_{54} = 2.00000154351297
x55=2.00000144989812x_{55} = 2.00000144989812
x56=2.00000145847506x_{56} = 2.00000145847506
x57=2.00000169888093x_{57} = 2.00000169888093
x58=2.00000132073716x_{58} = 2.00000132073716
x59=2.00000149578759x_{59} = 2.00000149578759
x60=2.00000121296052x_{60} = 2.00000121296052
x61=2.00000159646162x_{61} = 2.00000159646162
x62=2.00000133554685x_{62} = 2.00000133554685
x63=2.00000122505895x_{63} = 2.00000122505895
x64=2.00000079943602x_{64} = 2.00000079943602
x65=2.00000142591196x_{65} = 2.00000142591196
x66=2.00000133637072x_{66} = 2.00000133637072
x67=2.00000130737036x_{67} = 2.00000130737036
x68=2.00000151116422x_{68} = 2.00000151116422
x69=2.00000143553503x_{69} = 2.00000143553503
x70=2.00000143902485x_{70} = 2.00000143902485
x71=2.00000118189999x_{71} = 2.00000118189999
x72=2.00000142961767x_{72} = 2.00000142961767
x73=2.00000131914966x_{73} = 2.00000131914966
x74=2.00000148935129x_{74} = 2.00000148935129
x75=2.00000153112331x_{75} = 2.00000153112331
x76=2.00000124459547x_{76} = 2.00000124459547
x77=2.0000015204518x_{77} = 2.0000015204518
x78=2.00000133286299x_{78} = 2.00000133286299
x79=2.00000146180229x_{79} = 2.00000146180229
x80=2.00000128571475x_{80} = 2.00000128571475
x81=2.00000145254858x_{81} = 2.00000145254858
x82=2.0000012660126x_{82} = 2.0000012660126
x83=2.00000133188985x_{83} = 2.00000133188985
x84=2.00000142171605x_{84} = 2.00000142171605
x85=2.00000133468906x_{85} = 2.00000133468906
x86=2.00000131179417x_{86} = 2.00000131179417
x87=2.00000144511992x_{87} = 2.00000144511992
x88=2.00000144295891x_{88} = 2.00000144295891
x89=2.00000129316084x_{89} = 2.00000129316084
x90=2.00000155807206x_{90} = 2.00000155807206
x91=2.00000132630469x_{91} = 2.00000132630469
x92=2.00000143241823x_{92} = 2.00000143241823
x93=2.00000143393384x_{93} = 2.00000143393384
x94=2.00000132501832x_{94} = 2.00000132501832
x95=2.00000106309469x_{95} = 2.00000106309469
x96=2.00000127682118x_{96} = 2.00000127682118
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 2)^2/(x + 1).
(−2)21\frac{\left(-2\right)^{2}}{1}
The result:
f(0)=4f{\left(0 \right)} = 4
The point:
(0, 4)
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
−(x−2)2(x+1)2+2x−4x+1=0- \frac{\left(x - 2\right)^{2}}{\left(x + 1\right)^{2}} + \frac{2 x - 4}{x + 1} = 0
Solve this equation
The roots of this equation
x1=−4x_{1} = -4
x2=2x_{2} = 2
The values of the extrema at the points:
(-4, -12)

(2, 0)


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=2x_{1} = 2
Maxima of the function at points:
x1=−4x_{1} = -4
Decreasing at intervals
(−∞,−4]∪[2,∞)\left(-\infty, -4\right] \cup \left[2, \infty\right)
Increasing at intervals
[−4,2]\left[-4, 2\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((x−2)2(x+1)2−2(x−2)x+1+1)x+1=0\frac{2 \left(\frac{\left(x - 2\right)^{2}}{\left(x + 1\right)^{2}} - \frac{2 \left(x - 2\right)}{x + 1} + 1\right)}{x + 1} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=−1x_{1} = -1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x−2)2x+1)=−∞\lim_{x \to -\infty}\left(\frac{\left(x - 2\right)^{2}}{x + 1}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x−2)2x+1)=∞\lim_{x \to \infty}\left(\frac{\left(x - 2\right)^{2}}{x + 1}\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 - 2)^2/(x + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−2)2x(x+1))=1\lim_{x \to -\infty}\left(\frac{\left(x - 2\right)^{2}}{x \left(x + 1\right)}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞((x−2)2x(x+1))=1\lim_{x \to \infty}\left(\frac{\left(x - 2\right)^{2}}{x \left(x + 1\right)}\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:
(x−2)2x+1=(−x−2)21−x\frac{\left(x - 2\right)^{2}}{x + 1} = \frac{\left(- x - 2\right)^{2}}{1 - x}
- No
(x−2)2x+1=−(−x−2)21−x\frac{\left(x - 2\right)^{2}}{x + 1} = - \frac{\left(- x - 2\right)^{2}}{1 - x}
- No
so, the function
not is
neither even, nor odd
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