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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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              2
       (x + 1) 
f(x) = --------
        x - 2
f(x)=(x+1)2x−2f{\left(x \right)} = \frac{\left(x + 1\right)^{2}}{x - 2} 
f = (x + 1)^2/(x - 2)
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=2x_{1} = 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:
(x+1)2x−2=0\frac{\left(x + 1\right)^{2}}{x - 2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=−1.00000129752367x_{1} = -1.00000129752367
x2=−1.00000124747317x_{2} = -1.00000124747317
x3=−1.00000141930909x_{3} = -1.00000141930909
x4=−1.00000145079248x_{4} = -1.00000145079248
x5=−1.00000123877856x_{5} = -1.00000123877856
x6=−1.00000133318956x_{6} = -1.00000133318956
x7=−1.00000142109737x_{7} = -1.00000142109737
x8=−1.00000171717037x_{8} = -1.00000171717037
x9=−1.00000111645089x_{9} = -1.00000111645089
x10=−1.00000114567948x_{10} = -1.00000114567948
x11=−0.999999154657544x_{11} = -0.999999154657544
x12=−1.00000131446785x_{12} = -1.00000131446785
x13=−1.00000128709387x_{13} = -1.00000128709387
x14=−1.00000156371314x_{14} = -1.00000156371314
x15=−1.00000142517208x_{15} = -1.00000142517208
x16=−1.00000143146935x_{16} = -1.00000143146935
x17=−1.00000121735217x_{17} = -1.00000121735217
x18=−1.00000132908671x_{18} = -1.00000132908671
x19=−1.00000125514016x_{19} = -1.00000125514016
x20=−1.00000131250222x_{20} = -1.00000131250222
x21=−1.00000149818648x_{21} = -1.00000149818648
x22=−1.00000132274102x_{22} = -1.00000132274102
x23=−1.00000143008097x_{23} = -1.00000143008097
x24=−1.00000132126564x_{24} = -1.00000132126564
x25=−1.00000147522851x_{25} = -1.00000147522851
x26=−1.00000188396895x_{26} = -1.00000188396895
x27=−1.00000144826207x_{27} = -1.00000144826207
x28=−1.00000143967045x_{28} = -1.00000143967045
x29=−1.00000142018463x_{29} = -1.00000142018463
x30=−1.00000130817681x_{30} = -1.00000130817681
x31=−1.00000143293262x_{31} = -1.00000143293262
x32=−1.0000006697916x_{32} = -1.0000006697916
x33=−1.00000143610945x_{33} = -1.00000143610945
x34=−1.00000129086113x_{34} = -1.00000129086113
x35=−1.00000142750702x_{35} = -1.00000142750702
x36=−1.00000132414058x_{36} = -1.00000132414058
x37=−1.00000133664784x_{37} = -1.00000133664784
x38=−1.00000133018248x_{38} = -1.00000133018248
x39=−1.0000015239673x_{39} = -1.0000015239673
x40=−1.0000012288349x_{40} = -1.0000012288349
x41=−1.00000148011037x_{41} = -1.00000148011037
x42=−1.00000130048373x_{42} = -1.00000130048373
x43=−1.00000131040946x_{43} = -1.00000131040946
x44=−1.00000126195155x_{44} = -1.00000126195155
x45=−1.00000132673451x_{45} = -1.00000132673451
x46=−1.00000133498939x_{46} = -1.00000133498939
x47=−1.000001278479x_{47} = -1.000001278479
x48=−1.00000144161747x_{48} = -1.00000144161747
x49=−1.00000148550404x_{49} = -1.00000148550404
x50=−1.000001268043x_{50} = -1.000001268043
x51=−1.00000107877543x_{51} = -1.00000107877543
x52=−1.00000128298292x_{52} = -1.00000128298292
x53=−1.00000085038662x_{53} = -1.00000085038662
x54=−1.00000133122947x_{54} = -1.00000133122947
x55=−1.00000151423371x_{55} = -1.00000151423371
x56=−1.00000166912986x_{56} = -1.00000166912986
x57=−1.00000130323177x_{57} = -1.00000130323177
x58=−1.0000014263118x_{58} = -1.0000014263118
x59=−1.00000146301607x_{59} = -1.00000146301607
x60=−1.00000160480703x_{60} = -1.00000160480703
x61=−1.0000013180614x_{61} = -1.0000013180614
x62=−1.0000017841392x_{62} = -1.0000017841392
x63=−1.00000133583522x_{63} = -1.00000133583522
x64=−1.00000102837189x_{64} = -1.00000102837189
x65=−1.00000158222097x_{65} = -1.00000158222097
x66=−1.00000127352292x_{66} = -1.00000127352292
x67=−1.00000120394308x_{67} = -1.00000120394308
x68=−1.00000131970812x_{68} = -1.00000131970812
x69=−1.00000146673407x_{69} = -1.00000146673407
x70=−1.00000132547004x_{70} = -1.00000132547004
x71=−1.0000014378377x_{71} = -1.0000014378377
x72=−1.00000133223085x_{72} = -1.00000133223085
x73=−1.0000015351896x_{73} = -1.0000015351896
x74=−1.00000029822544x_{74} = -1.00000029822544
x75=−1.00000118807845x_{75} = -1.00000118807845
x76=−1.00000150571101x_{76} = -1.00000150571101
x77=−1.00000142304433x_{77} = -1.00000142304433
x78=−1.0000014535098x_{78} = -1.0000014535098
x79=−1.00000163298623x_{79} = -1.00000163298623
x80=−1.00000095747583x_{80} = -1.00000095747583
x81=−1.00000147078891x_{81} = -1.00000147078891
x82=−1.00000154827042x_{82} = -1.00000154827042
x83=−1.00000142408406x_{83} = -1.00000142408406
x84=−1.00000132793867x_{84} = -1.00000132793867
x85=−1.00000129432607x_{85} = -1.00000129432607
x86=−1.0000014436898x_{86} = -1.0000014436898
x87=−1.00000130578978x_{87} = -1.00000130578978
x88=−1.00000145643552x_{88} = -1.00000145643552
x89=−1.00000144589993x_{89} = -1.00000144589993
x90=−1.00000116901575x_{90} = -1.00000116901575
x91=−1.00000133410825x_{91} = -1.00000133410825
x92=−1.00000131631759x_{92} = -1.00000131631759
x93=−1.00000142204972x_{93} = -1.00000142204972
x94=−1.00000149149447x_{94} = -1.00000149149447
x95=−1.00000145959459x_{95} = -1.00000145959459
x96=−1.00000142876188x_{96} = -1.00000142876188
x97=−1.000001434477x_{97} = -1.000001434477
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x + 1)^2/(x - 2).
12−2\frac{1^{2}}{-2}
The result:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
The point:
(0, -1/2)
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
2x+2x−2−(x+1)2(x−2)2=0\frac{2 x + 2}{x - 2} - \frac{\left(x + 1\right)^{2}}{\left(x - 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
x2=5x_{2} = 5
The values of the extrema at the points:
(-1, 0)

(5, 12)


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=5x_{1} = 5
Maxima of the function at points:
x1=−1x_{1} = -1
Decreasing at intervals
(−∞,−1]∪[5,∞)\left(-\infty, -1\right] \cup \left[5, \infty\right)
Increasing at intervals
[−1,5]\left[-1, 5\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(1−2(x+1)x−2+(x+1)2(x−2)2)x−2=0\frac{2 \left(1 - \frac{2 \left(x + 1\right)}{x - 2} + \frac{\left(x + 1\right)^{2}}{\left(x - 2\right)^{2}}\right)}{x - 2} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=2x_{1} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x+1)2x−2)=−∞\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{2}}{x - 2}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x+1)2x−2)=∞\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{2}}{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 + 1)^2/(x - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x+1)2x(x−2))=1\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{2}}{x \left(x - 2\right)}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞((x+1)2x(x−2))=1\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{2}}{x \left(x - 2\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+1)2x−2=(1−x)2−x−2\frac{\left(x + 1\right)^{2}}{x - 2} = \frac{\left(1 - x\right)^{2}}{- x - 2}
- No
(x+1)2x−2=−(1−x)2−x−2\frac{\left(x + 1\right)^{2}}{x - 2} = - \frac{\left(1 - x\right)^{2}}{- x - 2}
- No
so, the function
not is
neither even, nor odd
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