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  • Graphing y =:
  • 2x^4-4x^2+1
  • е^(1/2-x)
  • y=(3,5|x|-1)/(|x|-3,5x^2)
  • x/(x²-4)

  • Identical expressions

  • exp(x)*(x^ two -x)
  • exponent of (x) multiply by (x squared minus x)
  • exponent of (x) multiply by (x to the power of two minus x)
  • exp(x)*(x2-x)
  • expx*x2-x
  • exp(x)*(x²-x)
  • exp(x)*(x to the power of 2-x)
  • exp(x)(x^2-x)
  • exp(x)(x2-x)
  • expxx2-x
  • expxx^2-x

  • Similar expressions

  • exp(x)*(x^2+x)
Plotting a function graph/ exp(x)*(x^2-x)

Graphing y = exp(x)*(x^2-x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        x / 2    \
f(x) = e *\x  - x/
f(x)=(x2−x)exf{\left(x \right)} = \left(x^{2} - x\right) e^{x} 
f = (x^2 - x)*exp(x)
The graph of the function
02468-8-6-4-2-10104000000-2000000
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:
(x2−x)ex=0\left(x^{2} - x\right) e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=1x_{2} = 1
Numerical solution
x1=−51.9820277550494x_{1} = -51.9820277550494
x2=−103.338185586309x_{2} = -103.338185586309
x3=1x_{3} = 1
x4=−81.4892524562269x_{4} = -81.4892524562269
x5=−95.3837003021592x_{5} = -95.3837003021592
x6=−109.309041253055x_{6} = -109.309041253055
x7=−91.4099726881485x_{7} = -91.4099726881485
x8=−40.5139335670481x_{8} = -40.5139335670481
x9=−38.6588105659299x_{9} = -38.6588105659299
x10=−121.26048557894x_{10} = -121.26048557894
x11=−75.5491881156761x_{11} = -75.5491881156761
x12=−79.5080107464477x_{12} = -79.5080107464477
x13=−36.8332858955167x_{13} = -36.8332858955167
x14=−65.6801741392534x_{14} = -65.6801741392534
x15=−57.8285952189519x_{15} = -57.8285952189519
x16=−107.31834162524x_{16} = -107.31834162524
x17=−61.7480858408277x_{17} = -61.7480858408277
x18=−59.7865542399166x_{18} = -59.7865542399166
x19=−99.3598809506193x_{19} = -99.3598809506193
x20=−119.267813516739x_{20} = -119.267813516739
x21=−77.5279506631343x_{21} = -77.5279506631343
x22=−44.2864913755559x_{22} = -44.2864913755559
x23=−97.3715060453445x_{23} = -97.3715060453445
x24=−101.348785977816x_{24} = -101.348785977816
x25=−111.300121308339x_{25} = -111.300121308339
x26=−89.4241507425458x_{26} = -89.4241507425458
x27=−48.1155320563989x_{27} = -48.1155320563989
x28=−69.6220986676911x_{28} = -69.6220986676911
x29=−83.4715733483481x_{29} = -83.4715733483481
x30=−67.6500464899076x_{30} = -67.6500464899076
x31=−50.0449269518681x_{31} = -50.0449269518681
x32=−85.4548825357727x_{32} = -85.4548825357727
x33=−42.3914836119009x_{33} = -42.3914836119009
x34=−115.283332833526x_{34} = -115.283332833526
x35=−46.1953873041686x_{35} = -46.1953873041686
x36=0x_{36} = 0
x37=−105.328047350538x_{37} = -105.328047350538
x38=−87.4390990971913x_{38} = -87.4390990971913
x39=1.00000000000032x_{39} = 1.00000000000032
x40=−63.7127493082472x_{40} = -63.7127493082472
x41=−73.5718547803295x_{41} = -73.5718547803295
x42=−35.0481976660927x_{42} = -35.0481976660927
x43=−53.9256206828219x_{43} = -53.9256206828219
x44=−93.3965067038405x_{44} = -93.3965067038405
x45=−55.8747373925463x_{45} = -55.8747373925463
x46=−117.275423715019x_{46} = -117.275423715019
x47=−113.291558872075x_{47} = -113.291558872075
x48=−71.5961008938983x_{48} = -71.5961008938983
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to exp(x)*(x^2 - x).
(02−0)e0\left(0^{2} - 0\right) e^{0}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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−1)ex+(x2−x)ex=0\left(2 x - 1\right) e^{x} + \left(x^{2} - x\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−12+52x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}
x2=−52−12x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}
The values of the extrema at the points:
                                                    ___ 
              /                 2        \    1   \/ 5  
         ___  |    /        ___\      ___|  - - + ----- 
   1   \/ 5   |1   |  1   \/ 5 |    \/ 5 |    2     2   
(- - + -----, |- + |- - + -----|  - -----|*e           )
   2     2    \2   \  2     2  /      2  /              

                                                    ___ 
              /                 2        \    1   \/ 5  
         ___  |    /        ___\      ___|  - - - ----- 
   1   \/ 5   |1   |  1   \/ 5 |    \/ 5 |    2     2   
(- - - -----, |- + |- - - -----|  + -----|*e           )
   2     2    \2   \  2     2  /      2  /


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=−12+52x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}
Maxima of the function at points:
x1=−52−12x_{1} = - \frac{\sqrt{5}}{2} - \frac{1}{2}
Decreasing at intervals
(−∞,−52−12]∪[−12+52,∞)\left(-\infty, - \frac{\sqrt{5}}{2} - \frac{1}{2}\right] \cup \left[- \frac{1}{2} + \frac{\sqrt{5}}{2}, \infty\right)
Increasing at intervals
[−52−12,−12+52]\left[- \frac{\sqrt{5}}{2} - \frac{1}{2}, - \frac{1}{2} + \frac{\sqrt{5}}{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
(x(x−1)+4x)ex=0\left(x \left(x - 1\right) + 4 x\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−3x_{1} = -3
x2=0x_{2} = 0

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,−3]∪[0,∞)\left(-\infty, -3\right] \cup \left[0, \infty\right)
Convex at the intervals
[−3,0]\left[-3, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x2−x)ex)=0\lim_{x \to -\infty}\left(\left(x^{2} - x\right) e^{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞((x2−x)ex)=∞\lim_{x \to \infty}\left(\left(x^{2} - x\right) e^{x}\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 exp(x)*(x^2 - x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x2−x)exx)=0\lim_{x \to -\infty}\left(\frac{\left(x^{2} - x\right) e^{x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((x2−x)exx)=∞\lim_{x \to \infty}\left(\frac{\left(x^{2} - x\right) e^{x}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
(x2−x)ex=(x2+x)e−x\left(x^{2} - x\right) e^{x} = \left(x^{2} + x\right) e^{- x}
- No
(x2−x)ex=−(x2+x)e−x\left(x^{2} - x\right) e^{x} = - \left(x^{2} + x\right) e^{- x}
- No
so, the function
not is
neither even, nor odd
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