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  • Graphing y =:
  • -x^2+2x+4
  • x^2+2x+4
  • -x^2-2x+3
  • x^2+14x+15

  • Identical expressions

  • ((x- two)^ two)/(x- one)
  • ((x minus 2) squared ) divide by (x minus 1)
  • ((x minus two) to the power of two) divide by (x minus one)
  • ((x-2)2)/(x-1)
  • x-22/x-1
  • ((x-2)²)/(x-1)
  • ((x-2) to the power of 2)/(x-1)
  • x-2^2/x-1
  • ((x-2)^2) divide by (x-1)

  • Similar expressions

  • ((x-2)^2)/(x+1)
  • ((x+2)^2)/(x-1)
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

You have entered [src] 
              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
0-80-60-40-2020406080-100100-250250
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=2.00000047916709x_{1} = 2.00000047916709
x2=2.00000100705804x_{2} = 2.00000100705804
x3=2.00000044933635x_{3} = 2.00000044933635
x4=2.00000046401639x_{4} = 2.00000046401639
x5=2.00000045368084x_{5} = 2.00000045368084
x6=2.00000051097964x_{6} = 2.00000051097964
x7=2.00000044332544x_{7} = 2.00000044332544
x8=2.00000046391212x_{8} = 2.00000046391212
x9=2.00000043510409x_{9} = 2.00000043510409
x10=2.00000052615216x_{10} = 2.00000052615216
x11=2.00000045355118x_{11} = 2.00000045355118
x12=2.00000046362414x_{12} = 2.00000046362414
x13=2.00000044791696x_{13} = 2.00000044791696
x14=2.00000045392272x_{14} = 2.00000045392272
x15=2.00000045444203x_{15} = 2.00000045444203
x16=2.00000046504588x_{16} = 2.00000046504588
x17=2.00000048109522x_{17} = 2.00000048109522
x18=2.0000004504353x_{18} = 2.0000004504353
x19=2.00000048343265x_{19} = 2.00000048343265
x20=2.00000046489434x_{20} = 2.00000046489434
x21=2.00000046847041x_{21} = 2.00000046847041
x22=2.00000045312185x_{22} = 2.00000045312185
x23=2.00000044671601x_{23} = 2.00000044671601
x24=2.00000045243353x_{24} = 2.00000045243353
x25=2.00000045202598x_{25} = 2.00000045202598
x26=2.00000044216563x_{26} = 2.00000044216563
x27=2.0000004542474x_{27} = 2.0000004542474
x28=2.00000045279651x_{28} = 2.00000045279651
x29=2.0000004644826x_{29} = 2.0000004644826
x30=2.000000464239x_{30} = 2.000000464239
x31=2.00000047754941x_{31} = 2.00000047754941
x32=2.00000046661682x_{32} = 2.00000046661682
x33=2.00000062061962x_{33} = 2.00000062061962
x34=2.00000044734716x_{34} = 2.00000044734716
x35=2.00000040891481x_{35} = 2.00000040891481
x36=2.00000048632517x_{36} = 2.00000048632517
x37=2.00000041791077x_{37} = 2.00000041791077
x38=2.00000046714773x_{38} = 2.00000046714773
x39=2.00000055386291x_{39} = 2.00000055386291
x40=2.00000046775901x_{40} = 2.00000046775901
x41=2.00000044843395x_{41} = 2.00000044843395
x42=2.00000031321749x_{42} = 2.00000031321749
x43=2.00000044973248x_{43} = 2.00000044973248
x44=2.0000004663767x_{44} = 2.0000004663767
x45=2.00000045341512x_{45} = 2.00000045341512
x46=2.00000050140513x_{46} = 2.00000050140513
x47=2.00000042417021x_{47} = 2.00000042417021
x48=2.00000046555183x_{48} = 2.00000046555183
x49=2.00000046687325x_{49} = 2.00000046687325
x50=2.00000043737024x_{50} = 2.00000043737024
x51=2.0000004661514x_{51} = 2.0000004661514
x52=2.00000046887168x_{52} = 2.00000046887168
x53=2.00000045327219x_{53} = 2.00000045327219
x54=2.00000042877704x_{54} = 2.00000042877704
x55=2.00000044890513x_{55} = 2.00000044890513
x56=2.00000046593959x_{56} = 2.00000046593959
x57=2.00000045009763x_{57} = 2.00000045009763
x58=2.00000044601304x_{58} = 2.00000044601304
x59=2.00000046371625x_{59} = 2.00000046371625
x60=2.00000045103974x_{60} = 2.00000045103974
x61=2.00000045131131x_{61} = 2.00000045131131
x62=2.00000046520552x_{62} = 2.00000046520552
x63=2.00000043924486x_{63} = 2.00000043924486
x64=2.0000004647503x_{64} = 2.0000004647503
x65=2.00000039488476x_{65} = 2.00000039488476
x66=2.00000046744222x_{66} = 2.00000046744222
x67=2.00000047617277x_{67} = 2.00000047617277
x68=2.00000047304881x_{68} = 2.00000047304881
x69=2.00000045223582x_{69} = 2.00000045223582
x70=2.00000046574007x_{70} = 2.00000046574007
x71=2.00000047153142x_{71} = 2.00000047153142
x72=2.00000047498704x_{72} = 2.00000047498704
x73=2.00000046930873x_{73} = 2.00000046930873
x74=2.00000036994737x_{74} = 2.00000036994737
x75=2.00000046381216x_{75} = 2.00000046381216
x76=2.00000045434672x_{76} = 2.00000045434672
x77=2.00000049481285x_{77} = 2.00000049481285
x78=2.00000047224656x_{78} = 2.00000047224656
x79=2.00000047395508x_{79} = 2.00000047395508
x80=2.00000044522527x_{80} = 2.00000044522527
x81=2.00000045262012x_{81} = 2.00000045262012
x82=2.00000044082137x_{82} = 2.00000044082137
x83=2.0000004540357x_{83} = 2.0000004540357
x84=2.00000045156512x_{84} = 2.00000045156512
x85=2.00000046978658x_{85} = 2.00000046978658
x86=2.00000046435799x_{86} = 2.00000046435799
x87=2.00000047088992x_{87} = 2.00000047088992
x88=2.00000045296351x_{88} = 2.00000045296351
x89=2.00000045414382x_{89} = 2.00000045414382
x90=2.00000046412524x_{90} = 2.00000046412524
x91=2x_{91} = 2
x92=2.00000045380456x_{92} = 2.00000045380456
x93=2.00000044433634x_{93} = 2.00000044433634
x94=2.00000045074848x_{94} = 2.00000045074848
x95=2.00000048999717x_{95} = 2.00000048999717
x96=2.00000045180284x_{96} = 2.00000045180284
x97=2.00000046810072x_{97} = 2.00000046810072
x98=2.00000046537392x_{98} = 2.00000046537392
x99=2.00000047031124x_{99} = 2.00000047031124
x100=2.00000046461322x_{100} = 2.00000046461322
x101=2.00000043230946x_{101} = 2.00000043230946
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)2−1\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=0x_{1} = 0
x2=2x_{2} = 2
The values of the extrema at the points:
(0, -4)

(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=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[2,∞)\left(-\infty, 0\right] \cup \left[2, \infty\right)
Increasing at intervals
[0,2]\left[0, 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)2−x−1\frac{\left(x - 2\right)^{2}}{x - 1} = \frac{\left(- x - 2\right)^{2}}{- x - 1}
- No
(x−2)2x−1=−(−x−2)2−x−1\frac{\left(x - 2\right)^{2}}{x - 1} = - \frac{\left(- x - 2\right)^{2}}{- x - 1}
- No
so, the function
not is
neither even, nor odd
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