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  • Graphing y =:
  • (x+2)^2/(x+1)
  • x^2-3x+3
  • -x^2-3
  • -x^2+2x+15

  • Identical expressions

  • (x+ two)^ two /(x+ one)
  • (x plus 2) squared divide by (x plus 1)
  • (x plus two) to the power of two divide by (x plus 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
02468-8-6-4-2-1010-5050
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=−2.00000046379709x_{1} = -2.00000046379709
x2=−2.00000045070158x_{2} = -2.00000045070158
x3=−2.00000047079664x_{3} = -2.00000047079664
x4=−2.00000046487161x_{4} = -2.00000046487161
x5=−2.00000029729834x_{5} = -2.00000029729834
x6=−2.00000045324946x_{6} = -2.00000045324946
x7=−2.00000043897666x_{7} = -2.00000043897666
x8=−2.00000044197101x_{8} = -2.00000044197101
x9=−2.0000006047672x_{9} = -2.0000006047672
x10=−2.00000046399998x_{10} = -2.00000046399998
x11=−2.00000044835722x_{11} = -2.00000044835722
x12=−2.00000044589723x_{12} = -2.00000044589723
x13=−2.00000045442756x_{13} = -2.00000045442756
x14=−2.00000046552369x_{14} = -2.00000046552369
x15=−2.00000045240373x_{15} = -2.00000045240373
x16=−2.00000046410811x_{16} = -2.00000046410811
x17=−2.00000045309793x_{17} = -2.00000045309793
x18=−2.00000050922702x_{18} = -2.00000050922702
x19=−2.00000045276989x_{19} = -2.00000045276989
x20=−2.0000004668325x_{20} = -2.0000004668325
x21=−2.00000047597816x_{21} = -2.00000047597816
x22=−2.00000042333046x_{22} = -2.00000042333046
x23=−2.00000043704851x_{23} = -2.00000043704851
x24=−2.00000042814636x_{24} = -2.00000042814636
x25=−2.0000004519924x_{25} = -2.0000004519924
x26=−2.0000004653473x_{26} = -2.0000004653473
x27=−2.00000047022684x_{27} = -2.00000047022684
x28=−2.0000004638964x_{28} = -2.0000004638964
x29=−2.00000044783257x_{29} = -2.00000044783257
x30=−2.00000085914622x_{30} = -2.00000085914622
x31=−2.00000045433164x_{31} = -2.00000045433164
x32=−2.00000047380746x_{32} = -2.00000047380746
x33=−2.00000048583419x_{33} = -2.00000048583419
x34=−2.00000047481835x_{34} = -2.00000047481835
x35=−2.00000046634096x_{35} = -2.00000046634096
x36=−2.00000047142779x_{36} = -2.00000047142779
x37=−2.00000045259198x_{37} = -2.00000045259198
x38=−2.00000036425585x_{38} = -2.00000036425585
x39=−2.00000045220422x_{39} = -2.00000045220422
x40=−2.00000041673771x_{40} = -2.00000041673771
x41=−2.00000046571028x_{41} = -2.00000046571028
x42=−2.00000045353059x_{42} = -2.00000045353059
x43=−2.00000045038479x_{43} = -2.00000045038479
x44=−2.00000046611783x_{44} = -2.00000046611783
x45=−2.00000045127056x_{45} = -2.00000045127056
x46=−2.00000045099608x_{46} = -2.00000045099608
x47=−2.00000046361024x_{47} = -2.00000046361024
x48=−2.00000047291853x_{48} = -2.00000047291853
x49=−2.00000046804618x_{49} = -2.00000046804618
x50=−2.00000048303961x_{50} = -2.00000048303961
x51=−2.00000044883508x_{51} = -2.00000044883508
x52=−2.00000044059437x_{52} = -2.00000044059437
x53=−2.00000046770851x_{53} = -2.00000046770851
x54=−2.00000047213075x_{54} = -2.00000047213075
x55=−2.00000040716196x_{55} = -2.00000040716196
x56=−2.0000004788989x_{56} = -2.0000004788989
x57=−2.00000045004308x_{57} = -2.00000045004308
x58=−2.00000039198545x_{58} = -2.00000039198545
x59=−2.00000045390481x_{59} = -2.00000045390481
x60=−2.00000046880745x_{60} = -2.00000046880745
x61=−2.00000046370178x_{61} = -2.00000046370178
x62=−2.00000046710406x_{62} = -2.00000046710406
x63=−2.00000044967339x_{63} = -2.00000044967339
x64=−2.00000045293829x_{64} = -2.00000045293829
x65=−2.00000044509499x_{65} = -2.00000044509499
x66=−2.00000046841133x_{66} = -2.00000046841133
x67=−2.00000047732241x_{67} = -2.00000047732241
x68=−2.00000046433925x_{68} = -2.00000046433925
x69=−2.0000004807735x_{69} = -2.0000004807735
x70=−2.0000004517671x_{70} = -2.0000004517671
x71=−2.00000050023215x_{71} = -2.00000050023215
x72=−2.00000045366121x_{72} = -2.00000045366121
x73=−2.00000046590798x_{73} = -2.00000046590798
x74=−2.00000044927213x_{74} = -2.00000044927213
x75=−2.0000004533935x_{75} = -2.0000004533935
x76=−2.00000045378581x_{76} = -2.00000045378581
x77=−2.00000046657869x_{77} = -2.00000046657869
x78=−2.00000046502195x_{78} = -2.00000046502195
x79=−2.00000046422109x_{79} = -2.00000046422109
x80=−2.00000044315675x_{80} = -2.00000044315675
x81=−2.00000046923867x_{81} = -2.00000046923867
x82=−2.00000046446296x_{82} = -2.00000046446296
x83=−2.00000052325369x_{83} = -2.00000052325369
x84=−2.00000046459263x_{84} = -2.00000046459263
x85=−2.00000045423169x_{85} = -2.00000045423169
x86=−2.00000046739533x_{86} = -2.00000046739533
x87=−2.00000043181848x_{87} = -2.00000043181848
x88=−2.00000043471105x_{88} = -2.00000043471105
x89=−2.00000046970986x_{89} = -2.00000046970986
x90=−2.00000044418871x_{90} = -2.00000044418871
x91=−2.00000048936651x_{91} = -2.00000048936651
x92=−2.00000044725389x_{92} = -2.00000044725389
x93=−2.00000045412742x_{93} = -2.00000045412742
x94=−2.000000548176x_{94} = -2.000000548176
x95=−2.00000049397314x_{95} = -2.00000049397314
x96=−2.00000045401857x_{96} = -2.00000045401857
x97=−2.00000045152699x_{97} = -2.00000045152699
x98=−2.00000046472868x_{98} = -2.00000046472868
x99=−2x_{99} = -2
x100=−2.0000004651803x_{100} = -2.0000004651803
x101=−2.00000044661238x_{101} = -2.00000044661238
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).
221\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=0x_{2} = 0
The values of the extrema at the points:
(-2, 0)

(0, 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=0x_{1} = 0
Maxima of the function at points:
x1=−2x_{1} = -2
Decreasing at intervals
(−∞,−2]∪[0,∞)\left(-\infty, -2\right] \cup \left[0, \infty\right)
Increasing at intervals
[−2,0]\left[-2, 0\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)21−x\frac{\left(x + 2\right)^{2}}{x + 1} = \frac{\left(2 - x\right)^{2}}{1 - x}
- No
(x+2)2x+1=−(2−x)21−x\frac{\left(x + 2\right)^{2}}{x + 1} = - \frac{\left(2 - x\right)^{2}}{1 - x}
- No
so, the function
not is
neither even, nor odd
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