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Plotting a function graph/ (x-3)*sqrt(x-3)

Graphing y = (x-3)*sqrt(x-3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = (x - 3)*\/ x - 3
f(x)=x−3(x−3)f{\left(x \right)} = \sqrt{x - 3} \left(x - 3\right) 
f = sqrt(x - 3)*(x - 3)
The graph of the function
02468-8-6-4-2-1010020
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−3(x−3)=0\sqrt{x - 3} \left(x - 3\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3x_{1} = 3
Numerical solution
x1=2.99999999446763x_{1} = 2.99999999446763
x2=3.00000000811812x_{2} = 3.00000000811812
x3=2.99999999736838x_{3} = 2.99999999736838
x4=3.00000000282852x_{4} = 3.00000000282852
x5=2.99999999702711x_{5} = 2.99999999702711
x6=3.00000000248726x_{6} = 3.00000000248726
x7=2.99999999173752x_{7} = 2.99999999173752
x8=2.99999999753901x_{8} = 2.99999999753901
x9=3.00000000402295x_{9} = 3.00000000402295
x10=2.99999999310257x_{10} = 2.99999999310257
x11=3.00000000777686x_{11} = 3.00000000777686
x12=3.00000000538801x_{12} = 3.00000000538801
x13=3.00000000572927x_{13} = 3.00000000572927
x14=3.0000000064118x_{14} = 3.0000000064118
x15=2.99999999770965x_{15} = 2.99999999770965
x16=3.00000000453485x_{16} = 3.00000000453485
x17=2.99999999259067x_{17} = 2.99999999259067
x18=2.99999999463826x_{18} = 2.99999999463826
x19=2.99999999207878x_{19} = 2.99999999207878
x20=2.99999999788028x_{20} = 2.99999999788028
x21=3.00000000487611x_{21} = 3.00000000487611
x22=2.99999999583268x_{22} = 2.99999999583268
x23=3.00000000163408x_{23} = 3.00000000163408
x24=3.00000000334042x_{24} = 3.00000000334042
x25=2.99999999412636x_{25} = 2.99999999412636
x26=2.99999999276131x_{26} = 2.99999999276131
x27=3.00000000026874x_{27} = 3.00000000026874
x28=2.99999999242004x_{28} = 2.99999999242004
x29=2.9999999932732x_{29} = 2.9999999932732
x30=2.99999999293194x_{30} = 2.99999999293194
x31=3.00000000180472x_{31} = 3.00000000180472
x32=2.99999999839219x_{32} = 2.99999999839219
x33=3.00000000436422x_{33} = 3.00000000436422
x34=3.00000000112216x_{34} = 3.00000000112216
x35=3.00000000231662x_{35} = 3.00000000231662
x36=3.00000000521738x_{36} = 3.00000000521738
x37=2.99999999975755x_{37} = 2.99999999975755
x38=3.00000000299916x_{38} = 3.00000000299916
x39=2.99999999600332x_{39} = 2.99999999600332
x40=3.0000000000975x_{40} = 3.0000000000975
x41=2.99999999719775x_{41} = 2.99999999719775
x42=3.00000000470548x_{42} = 3.00000000470548
x43=2.99999999651521x_{43} = 2.99999999651521
x44=3.00000000794749x_{44} = 3.00000000794749
x45=2.99999999668585x_{45} = 2.99999999668585
x46=2.99999999156688x_{46} = 2.99999999156688
x47=3.00000000368169x_{47} = 3.00000000368169
x48=3.00000000351106x_{48} = 3.00000000351106
x49=2.99999999907476x_{49} = 2.99999999907476
x50=2.99999999480889x_{50} = 2.99999999480889
x51=2.99999999190815x_{51} = 2.99999999190815
x52=3.00000000385232x_{52} = 3.00000000385232
x53=3.00000000709433x_{53} = 3.00000000709433
x54=3.00000000214599x_{54} = 3.00000000214599
x55=2.99999999941608x_{55} = 2.99999999941608
x56=3.00000000726496x_{56} = 3.00000000726496
x57=3.0000000012928x_{57} = 3.0000000012928
x58=2.99999999992899x_{58} = 2.99999999992899
x59=2.99999999344383x_{59} = 2.99999999344383
x60=3.0000000069237x_{60} = 3.0000000069237
x61=3.00000000504675x_{61} = 3.00000000504675
x62=2.99999999617395x_{62} = 2.99999999617395
x63=3.00000000743559x_{63} = 3.00000000743559
x64=3.00000000265789x_{64} = 3.00000000265789
x65=3.00000000624117x_{65} = 3.00000000624117
x66=3.00000000555864x_{66} = 3.00000000555864
x67=2.99999999890411x_{67} = 2.99999999890411
x68=3.00000000828875x_{68} = 3.00000000828875
x69=3.0000000006102x_{69} = 3.0000000006102
x70=2.99999999634458x_{70} = 2.99999999634458
x71=2.99999999685648x_{71} = 2.99999999685648
x72=2.99999999924541x_{72} = 2.99999999924541
x73=2.99999999958677x_{73} = 2.99999999958677
x74=2.99999999532079x_{74} = 2.99999999532079
x75=3.00000000197535x_{75} = 3.00000000197535
x76=2.99999999566205x_{76} = 2.99999999566205
x77=2.9999999937851x_{77} = 2.9999999937851
x78=3.00000000146344x_{78} = 3.00000000146344
x79=3.00000000589991x_{79} = 3.00000000589991
x80=3.00000000760623x_{80} = 3.00000000760623
x81=2.99999999805092x_{81} = 2.99999999805092
x82=2.99999999361447x_{82} = 2.99999999361447
x83=2.99999999822155x_{83} = 2.99999999822155
x84=2.99999999515015x_{84} = 2.99999999515015
x85=2.99999999856283x_{85} = 2.99999999856283
x86=2.99999999497952x_{86} = 2.99999999497952
x87=2.99999999224941x_{87} = 2.99999999224941
x88=3.00000000419359x_{88} = 3.00000000419359
x89=2.99999999873347x_{89} = 2.99999999873347
x90=3.00000000607054x_{90} = 3.00000000607054
x91=3.00000000675307x_{91} = 3.00000000675307
x92=3.00000000043951x_{92} = 3.00000000043951
x93=2.99999999395573x_{93} = 2.99999999395573
x94=2.99999999549142x_{94} = 2.99999999549142
x95=3.00000000078086x_{95} = 3.00000000078086
x96=3.00000000095151x_{96} = 3.00000000095151
x97=2.99999999429699x_{97} = 2.99999999429699
x98=3.00000000658243x_{98} = 3.00000000658243
x99=2.99999999122562x_{99} = 2.99999999122562
x100=3.00000000316979x_{100} = 3.00000000316979
x101=2.99999999139625x_{101} = 2.99999999139625
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 3)*sqrt(x - 3).
−3−3- 3 \sqrt{-3}
The result:
f(0)=−33if{\left(0 \right)} = - 3 \sqrt{3} i
The point:
(0, -3*i*sqrt(3))
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
3x−32=0\frac{3 \sqrt{x - 3}}{2} = 0
Solve this equation
The roots of this equation
x1=3x_{1} = 3
The values of the extrema at the points:
(3, 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:
The function has no minima
The function has no maxima
Increasing at the entire real axis
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
34x−3=0\frac{3}{4 \sqrt{x - 3}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x−3(x−3))=−∞i\lim_{x \to -\infty}\left(\sqrt{x - 3} \left(x - 3\right)\right) = - \infty i
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x−3(x−3))=∞\lim_{x \to \infty}\left(\sqrt{x - 3} \left(x - 3\right)\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 - 3)*sqrt(x - 3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−3)32x)=∞i\lim_{x \to -\infty}\left(\frac{\left(x - 3\right)^{\frac{3}{2}}}{x}\right) = \infty i
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((x−3)32x)=∞\lim_{x \to \infty}\left(\frac{\left(x - 3\right)^{\frac{3}{2}}}{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:
x−3(x−3)=(−x−3)32\sqrt{x - 3} \left(x - 3\right) = \left(- x - 3\right)^{\frac{3}{2}}
- No
x−3(x−3)=−(−x−3)32\sqrt{x - 3} \left(x - 3\right) = - \left(- x - 3\right)^{\frac{3}{2}}
- No
so, the function
not is
neither even, nor odd
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