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Determine any asymptotes.
The function is not defined if
x 2 x 2 0
1x 221x 12 0
x 2 and x 1
The domain is 5xR 0 x 2 and x 16.
There are vertical asymptotes at x 2 and x 1.
Using f 1x2 x 4 we examine function values near the asymptotes.
x 2 x 2 ,
lim
xS1
lim
xS2
f 1x2 q
f 1x2 q
Sketch the information you have so far, as shown.
y
4
x = –1 x = 2
3
2
(0, 2)
1
(4, 0) x
–2 –1 0
–1
1 2 3 4
–2
lim f 1x2 q
xS1
lim
xS2
f 1x2 q
Analyze f ¿(x). Now determine the critical points.
f 1x2 1x 421x 2 x 22 1
f ¿1x2 1121x 2 x 22 1 1x 421121x 2 x 22 2 12x 12
1
x 2 x 2
1x2 x 22
1x 2 x 22 2 12x2 9x 42
1x 2 x 22 2
x2 8x 6
1x 2 x 22 2
f ¿1x2 0 if x 2 8x 6 0
x 8 ; 210
2
x 4 ; 10
1x 4212x 12
1x 2 x 22 2
Since we are sketching, approximate values 7.2 and 0.8 are acceptable. These
values give the approximate points 17.2, 0.12 and 10.8, 1.52.
210 4.5 AN ALGORITHM FOR CURVE SKETCHING
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