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((x-2)^2)e^(-x)
Plotting a function graph/ ((x-2)^2)e^(-x)

Graphing y = ((x-2)^2)e^(-x)

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The graph:

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Intersection points:

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Piecewise:

The solution

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              2  -x
f(x) = (x - 2) *e
f(x)=(x−2)2e−xf{\left(x \right)} = \left(x - 2\right)^{2} e^{- x} 
f = (x - 1*2)^2/E^x
The graph of the function
0102030405060708090100110-1005000000
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)2e−x=0\left(x - 2\right)^{2} e^{- x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2x_{1} = 2
Numerical solution
x1=64.0611807434853x_{1} = 64.0611807434853
x2=115.602013088993x_{2} = 115.602013088993
x3=46.6050925906729x_{3} = 46.6050925906729
x4=62.0999560358985x_{4} = 62.0999560358985
x5=60.1423474863896x_{5} = 60.1423474863896
x6=35.6579430943359x_{6} = 35.6579430943359
x7=42.5820728530031x_{7} = 42.5820728530031
x8=48.5128714785856x_{8} = 48.5128714785856
x9=50.4320998819442x_{9} = 50.4320998819442
x10=95.707404744577x_{10} = 95.707404744577
x11=99.6822895145426x_{11} = 99.6822895145426
x12=58.1888924840258x_{12} = 58.1888924840258
x13=39.1602455397125x_{13} = 39.1602455397125
x14=101.670613028306x_{14} = 101.670613028306
x15=52.3607330233137x_{15} = 52.3607330233137
x16=87.7660696193442x_{16} = 87.7660696193442
x17=85.7828486140689x_{17} = 85.7828486140689
x18=81.8194870788507x_{18} = 81.8194870788507
x19=2x_{19} = 2
x20=56.2402420845623x_{20} = 56.2402420845623
x21=37.379255492682x_{21} = 37.379255492682
x22=93.720934730719x_{22} = 93.720934730719
x23=119.585818346237x_{23} = 119.585818346237
x24=67.9927593677372x_{24} = 67.9927593677372
x25=66.0255739002577x_{25} = 66.0255739002577
x26=69.9624187188197x_{26} = 69.9624187188197
x27=71.9342805013838x_{27} = 71.9342805013838
x28=117.593756384128x_{28} = 117.593756384128
x29=83.8006238116621x_{29} = 83.8006238116621
x30=97.6945389638031x_{30} = 97.6945389638031
x31=109.628899840344x_{31} = 109.628899840344
x32=79.8395419968606x_{32} = 79.8395419968606
x33=105.648824952827x_{33} = 105.648824952827
x34=89.7502050583631x_{34} = 89.7502050583631
x35=54.2971932633301x_{35} = 54.2971932633301
x36=107.638644821409x_{36} = 107.638644821409
x37=103.659470122749x_{37} = 103.659470122749
x38=40.9827879874711x_{38} = 40.9827879874711
x39=77.8609058011359x_{39} = 77.8609058011359
x40=42.8356618339334x_{40} = 42.8356618339334
x41=44.7114678029016x_{41} = 44.7114678029016
x42=73.9081118282112x_{42} = 73.9081118282112
x43=113.610608082484x_{43} = 113.610608082484
x44=75.8837117221529x_{44} = 75.8837117221529
x45=91.7351819043081x_{45} = 91.7351819043081
x46=111.619562634492x_{46} = 111.619562634492
x47=121.578180845004x_{47} = 121.578180845004
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 1*2)^2/E^x.
((−1)2+0)2e(−1)0\left(\left(-1\right) 2 + 0\right)^{2} e^{\left(-1\right) 0}
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)2e−x+(2x−4)e−x=0- \left(x - 2\right)^{2} e^{- x} + \left(2 x - 4\right) e^{- x} = 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  -2 
(2, (-2 + 2) *e  )

            2  -4 
(4, (-2 + 4) *e  )


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=4x_{1} = 4
Decreasing at intervals
[2,4]\left[2, 4\right]
Increasing at intervals
(−∞,2]∪[4,∞)\left(-\infty, 2\right] \cup \left[4, \infty\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−2)2−4x+10)e−x=0\left(\left(x - 2\right)^{2} - 4 x + 10\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=−2+4x_{1} = - \sqrt{2} + 4
x2=2+4x_{2} = \sqrt{2} + 4

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
(−∞,−2+4]∪[2+4,∞)\left(-\infty, - \sqrt{2} + 4\right] \cup \left[\sqrt{2} + 4, \infty\right)
Convex at the intervals
[−2+4,2+4]\left[- \sqrt{2} + 4, \sqrt{2} + 4\right]
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)2e−x)=∞\lim_{x \to -\infty}\left(\left(x - 2\right)^{2} e^{- x}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x−2)2e−x)=0\lim_{x \to \infty}\left(\left(x - 2\right)^{2} e^{- x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x - 1*2)^2/E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−2)2e−xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(x - 2\right)^{2} e^{- x}}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((x−2)2e−xx)=0\lim_{x \to \infty}\left(\frac{\left(x - 2\right)^{2} e^{- x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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)2e−x=(−x−2)2ex\left(x - 2\right)^{2} e^{- x} = \left(- x - 2\right)^{2} e^{x}
- No
(x−2)2e−x=−(−x−2)2ex\left(x - 2\right)^{2} e^{- x} = - \left(- x - 2\right)^{2} e^{x}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = ((x-2)^2)e^(-x)
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