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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       x*x - 2*x + 1
f(x) = -------------
           x - 2
f(x)=(−2x+xx)+1x−2f{\left(x \right)} = \frac{\left(- 2 x + x x\right) + 1}{x - 2} 
f = (-2*x + x*x + 1)/(x - 2)
The graph of the function
02468-8-6-4-2-1010-5050
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:
(−2x+xx)+1x−2=0\frac{\left(- 2 x + x x\right) + 1}{x - 2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=0.999999327380525x_{1} = 0.999999327380525
x2=0.999999524622346x_{2} = 0.999999524622346
x3=0.999999530531072x_{3} = 0.999999530531072
x4=0.999999527486891x_{4} = 0.999999527486891
x5=0.999999535599611x_{5} = 0.999999535599611
x6=0.999999546037679x_{6} = 0.999999546037679
x7=0.999999528896054x_{7} = 0.999999528896054
x8=0.999999561943277x_{8} = 0.999999561943277
x9=0.999999553735217x_{9} = 0.999999553735217
x10=0.999999553058391x_{10} = 0.999999553058391
x11=0.99999957968753x_{11} = 0.99999957968753
x12=0.999999548351013x_{12} = 0.999999548351013
x13=0.999999520206113x_{13} = 0.999999520206113
x14=0.999999545926414x_{14} = 0.999999545926414
x15=0.999999547500797x_{15} = 0.999999547500797
x16=0.999999512508993x_{16} = 0.999999512508993
x17=0.999999546403323x_{17} = 0.999999546403323
x18=0.999999531394722x_{18} = 0.999999531394722
x19=0.999999587563591x_{19} = 0.999999587563591
x20=0.999999551398402x_{20} = 0.999999551398402
x21=0.999999536251054x_{21} = 0.999999536251054
x22=0.999999574087628x_{22} = 0.999999574087628
x23=0.999999525703861x_{23} = 0.999999525703861
x24=0.99999955534392x_{24} = 0.99999955534392
x25=0.99999946631169x_{25} = 0.99999946631169
x26=0.999999533542086x_{26} = 0.999999533542086
x27=0.999999791899624x_{27} = 0.999999791899624
x28=0.99999954698084x_{28} = 0.99999954698084
x29=0.999999536153772x_{29} = 0.999999536153772
x30=0.999999546154011x_{30} = 0.999999546154011
x31=0.999999532125894x_{31} = 0.999999532125894
x32=0.999999532752898x_{32} = 0.999999532752898
x33=0.999999545717814x_{33} = 0.999999545717814
x34=0.999999557414828x_{34} = 0.999999557414828
x35=0.999999515640072x_{35} = 0.999999515640072
x36=0.999999531774886x_{36} = 0.999999531774886
x37=0.999999530981609x_{37} = 0.999999530981609
x38=0.999999533988681x_{38} = 0.999999533988681
x39=0.999999430351275x_{39} = 0.999999430351275
x40=0.999999533034055x_{40} = 0.999999533034055
x41=0.99999954561991x_{41} = 0.99999954561991
x42=0.999999550138055x_{42} = 0.999999550138055
x43=0.999999532450974x_{43} = 0.999999532450974
x44=0.999999535054178x_{44} = 0.999999535054178
x45=0.999999546275763x_{45} = 0.999999546275763
x46=0.999999549148533x_{46} = 0.999999549148533
x47=0.999999558690677x_{47} = 0.999999558690677
x48=0.999999533296516x_{48} = 0.999999533296516
x49=0.999999508492931x_{49} = 0.999999508492931
x50=0.999999546825333x_{50} = 0.999999546825333
x51=0.999999545525929x_{51} = 0.999999545525929
x52=0.999999518149667x_{52} = 0.999999518149667
x53=0.99999954945387x_{53} = 0.99999954945387
x54=0.999999550941631x_{54} = 0.999999550941631
x55=0.999999530037782x_{55} = 0.999999530037782
x56=0.999999545435638x_{56} = 0.999999545435638
x57=0.99999952665083x_{57} = 0.99999952665083
x58=0.999999550523047x_{58} = 0.999999550523047
x59=0.999999535720501x_{59} = 0.999999535720501
x60=0.999999484620168x_{60} = 0.999999484620168
x61=0.999999548118484x_{61} = 0.999999548118484
x62=0.99999954790008x_{62} = 0.99999954790008
x63=0.999999660465295x_{63} = 0.999999660465295
x64=0.999999529495352x_{64} = 0.999999529495352
x65=0.999999495713475x_{65} = 0.999999495713475
x66=0.999999534737329x_{66} = 0.999999534737329
x67=0.999999549782766x_{67} = 0.999999549782766
x68=0.999999503154896x_{68} = 0.999999503154896
x69=0.999999545819891x_{69} = 0.999999545819891
x70=0.99999953594654x_{70} = 0.99999953594654
x71=0.999999556309983x_{71} = 0.999999556309983
x72=0.99999954886431x_{72} = 0.99999954886431
x73=0.999999619502919x_{73} = 0.999999619502919
x74=0.999999566654177x_{74} = 0.999999566654177
x75=0.999999523375418x_{75} = 0.999999523375418
x76=0.999999599458166x_{76} = 0.999999599458166
x77=0.999999546677616x_{77} = 0.999999546677616
x78=0.999999521921984x_{78} = 0.999999521921984
x79=0.999999535472975x_{79} = 0.999999535472975
x80=0.999999533772345x_{80} = 0.999999533772345
x81=0.999999536052357x_{81} = 0.999999536052357
x82=0.999999534899914x_{82} = 0.999999534899914
x83=0.999999547144771x_{83} = 0.999999547144771
x84=0.999999560180562x_{84} = 0.999999560180562
x85=0.999999546537118x_{85} = 0.999999546537118
x86=0.999999564061318x_{86} = 0.999999564061318
x87=0.99999953456573x_{87} = 0.99999953456573
x88=0.999999535340173x_{88} = 0.999999535340173
x89=0.999999569901687x_{89} = 0.999999569901687
x90=0.99999953419232x_{90} = 0.99999953419232
x91=0.999999535836028x_{91} = 0.999999535836028
x92=0.999999534384348x_{92} = 0.999999534384348
x93=0.999999547694554x_{93} = 0.999999547694554
x94=0.999999547317829x_{94} = 0.999999547317829
x95=0.99999952823045x_{95} = 0.99999952823045
x96=0.999999554492036x_{96} = 0.999999554492036
x97=0.999999551898838x_{97} = 0.999999551898838
x98=0.999999552449512x_{98} = 0.999999552449512
x99=0.999999535200744x_{99} = 0.999999535200744
x100=0.999999548599084x_{100} = 0.999999548599084
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x*x - 2*x + 1)/(x - 2).
(0⋅0−0)+1−2\frac{\left(0 \cdot 0 - 0\right) + 1}{-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−(−2x+xx)+1(x−2)2=0\frac{2 x - 2}{x - 2} - \frac{\left(- 2 x + x x\right) + 1}{\left(x - 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
x2=3x_{2} = 3
The values of the extrema at the points:
(1, 0)

(3, 4)


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=3x_{1} = 3
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
(−∞,1]∪[3,∞)\left(-\infty, 1\right] \cup \left[3, \infty\right)
Increasing at intervals
[1,3]\left[1, 3\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+x2−2x+1(x−2)2)x−2=0\frac{2 \left(1 - \frac{2 \left(x - 1\right)}{x - 2} + \frac{x^{2} - 2 x + 1}{\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→−∞((−2x+xx)+1x−2)=−∞\lim_{x \to -\infty}\left(\frac{\left(- 2 x + x x\right) + 1}{x - 2}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((−2x+xx)+1x−2)=∞\lim_{x \to \infty}\left(\frac{\left(- 2 x + x x\right) + 1}{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*x - 2*x + 1)/(x - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((−2x+xx)+1x(x−2))=1\lim_{x \to -\infty}\left(\frac{\left(- 2 x + x x\right) + 1}{x \left(x - 2\right)}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞((−2x+xx)+1x(x−2))=1\lim_{x \to \infty}\left(\frac{\left(- 2 x + x x\right) + 1}{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:
(−2x+xx)+1x−2=x2+2x+1−x−2\frac{\left(- 2 x + x x\right) + 1}{x - 2} = \frac{x^{2} + 2 x + 1}{- x - 2}
- No
(−2x+xx)+1x−2=−x2+2x+1−x−2\frac{\left(- 2 x + x x\right) + 1}{x - 2} = - \frac{x^{2} + 2 x + 1}{- x - 2}
- No
so, the function
not is
neither even, nor odd
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