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  • Graphing y =:
  • -x^2-2x+1
  • (x+1)(x-2)^2
  • 6x-2x^2
  • 9^(1/(x-3))

  • Identical expressions

  • (x+ two)^ two /(x- one)
  • (x plus 2) squared divide by (x minus 1)
  • (x plus 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
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=−2.00000141888457x_{1} = -2.00000141888457
x2=−2.00000142355836x_{2} = -2.00000142355836
x3=−2.00000127606163x_{3} = -2.00000127606163
x4=−2.00000150983557x_{4} = -2.00000150983557
x5=−2.00000127085349x_{5} = -2.00000127085349
x6=−2.00000131540656x_{6} = -2.00000131540656
x7=−2.00000149474529x_{7} = -2.00000149474529
x8=−2.00000145494514x_{8} = -2.00000145494514
x9=−2.00000128508423x_{9} = -2.00000128508423
x10=−2.00000119635802x_{10} = -2.00000119635802
x11=−2.00000130700358x_{11} = -2.00000130700358
x12=−2.00000132049757x_{12} = -2.00000132049757
x13=−2.00000129595631x_{13} = -2.00000129595631
x14=−2.00000132734386x_{14} = -2.00000132734386
x15=−2.00000195482268x_{15} = -2.00000195482268
x16=−2.0000016180752x_{16} = -2.0000016180752
x17=−2.0000014206362x_{17} = -2.0000014206362
x18=−2.00000129903171x_{18} = -2.00000129903171
x19=−2.00000147295727x_{19} = -2.00000147295727
x20=−2.00000142573525x_{20} = -2.00000142573525
x21=−2.00000130453327x_{21} = -2.00000130453327
x22=−2.00000132481374x_{22} = -2.00000132481374
x23=−2.00000128078352x_{23} = -2.00000128078352
x24=−2.00000169132379x_{24} = -2.00000169132379
x25=−2.00000142462183x_{25} = -2.00000142462183
x26=−2.00000155565829x_{26} = -2.00000155565829
x27=−2.00000129262907x_{27} = -2.00000129262907
x28=−2.00000131889638x_{28} = -2.00000131889638
x29=−2.00000125142338x_{29} = -2.00000125142338
x30=−2.00000144062902x_{30} = -2.00000144062902
x31=−2.0000012339791x_{31} = -2.0000012339791
x32=−2.00000144950514x_{32} = -2.00000144950514
x33=−2.00000150183527x_{33} = -2.00000150183527
x34=−2.00000090999285x_{34} = -2.00000090999285
x35=−2.00000133271537x_{35} = -2.00000133271537
x36=−2.00000146871659x_{36} = -2.00000146871659
x37=−2.00000146483565x_{37} = -2.00000146483565
x38=−2.00000143076616x_{38} = -2.00000143076616
x39=−2.00000128901771x_{39} = -2.00000128901771
x40=−2.00000148841864x_{40} = -2.00000148841864
x41=−2.00000164986538x_{41} = -2.00000164986538
x42=−2.00000143874043x_{42} = -2.00000143874043
x43=−2.00000142690222x_{43} = -2.00000142690222
x44=−2.00000157252958x_{44} = -2.00000157252958
x45=−2.00000132964091x_{45} = -2.00000132964091
x46=−2.00000142812672x_{46} = -2.00000142812672
x47=−2.00000132851945x_{47} = -2.00000132851945
x48=−2.00000126507994x_{48} = -2.00000126507994
x49=−2.00000133071187x_{49} = -2.00000133071187
x50=−2.00000159292437x_{50} = -2.00000159292437
x51=−2.00000122330742x_{51} = -2.00000122330742
x52=−2.0000014294131x_{52} = -2.0000014294131
x53=−2.00000151893356x_{53} = -2.00000151893356
x54=−2.00000147761012x_{54} = -2.00000147761012
x55=−2.00000144263723x_{55} = -2.00000144263723
x56=−2.00000130931148x_{56} = -2.00000130931148
x57=−2.0000012109175x_{57} = -2.0000012109175
x58=−2.00000142254157x_{58} = -2.00000142254157
x59=−2.0000013018828x_{59} = -2.0000013018828
x60=−2.00000115796687x_{60} = -2.00000115796687
x61=−2.00000109888248x_{61} = -2.00000109888248
x62=−2.00000132611009x_{62} = -2.00000132611009
x63=−2.00000154146998x_{63} = -2.00000154146998
x64=−2.0000015293718x_{64} = -2.0000015293718
x65=−2.00000052317146x_{65} = -2.00000052317146
x66=−2.00000113193293x_{66} = -2.00000113193293
x67=−2.00000124326681x_{67} = -2.00000124326681
x68=−2.00000133624556x_{68} = -2.00000133624556
x69=−2.00000132201318x_{69} = -2.00000132201318
x70=−2.00000131147249x_{70} = -2.00000131147249
x71=−2.00000174765223x_{71} = -2.00000174765223
x72=−2.00000145212651x_{72} = -2.00000145212651
x73=−2.00000182859988x_{73} = -2.00000182859988
x74=−2.00000131350019x_{74} = -2.00000131350019
x75=−2.00000131720216x_{75} = -2.00000131720216
x76=−2.0000011790044x_{76} = -2.0000011790044
x77=−2.00000077332479x_{77} = -2.00000077332479
x78=−2.00000143369425x_{78} = -2.00000143369425
x79=−2.00000143696108x_{79} = -2.00000143696108
x80=−2.00000144706103x_{80} = -2.00000144706103
x81=−2.00000143219125x_{81} = -2.00000143219125
x82=−2.00000133455337x_{82} = -2.00000133455337
x83=−2.00000133541658x_{83} = -2.00000133541658
x84=−2.00000144477679x_{84} = -2.00000144477679
x85=−2.00000143528175x_{85} = -2.00000143528175
x86=−2.00000125864359x_{86} = -2.00000125864359
x87=−2.00000142156843x_{87} = -2.00000142156843
x88=−2.00000133365375x_{88} = -2.00000133365375
x89=−2.00000146127052x_{89} = -2.00000146127052
x90=−2.00000148273833x_{90} = -2.00000148273833
x91=−2.00000145798418x_{91} = -2.00000145798418
x92=−2.00000105553648x_{92} = -2.00000105553648
x93=−2.00000141974236x_{93} = -2.00000141974236
x94=−2.00000133173567x_{94} = -2.00000133173567
x95=−2x_{95} = -2
x96=−2.00000099619474x_{96} = -2.00000099619474
x97=−2.00000132344991x_{97} = -2.00000132344991
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).
22−1\frac{2^{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
2x+4x−1−(x+2)2(x−1)2=0\frac{2 x + 4}{x - 1} - \frac{\left(x + 2\right)^{2}}{\left(x - 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−2x_{1} = -2
x2=4x_{2} = 4
The values of the extrema at the points:
(-2, 0)

(4, 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=4x_{1} = 4
Maxima of the function at points:
x1=−2x_{1} = -2
Decreasing at intervals
(−∞,−2]∪[4,∞)\left(-\infty, -2\right] \cup \left[4, \infty\right)
Increasing at intervals
[−2,4]\left[-2, 4\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+2)x−1+(x+2)2(x−1)2)x−1=0\frac{2 \left(1 - \frac{2 \left(x + 2\right)}{x - 1} + \frac{\left(x + 2\right)^{2}}{\left(x - 1\right)^{2}}\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=(2−x)2−x−1\frac{\left(x + 2\right)^{2}}{x - 1} = \frac{\left(2 - x\right)^{2}}{- x - 1}
- No
(x+2)2x−1=−(2−x)2−x−1\frac{\left(x + 2\right)^{2}}{x - 1} = - \frac{\left(2 - x\right)^{2}}{- x - 1}
- No
so, the function
not is
neither even, nor odd
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