Rational Function Graphing Question

[From: ] [author: ] [Date: 11-05-07] [Hit: ]

e.1 = (x^2) / (x^2 + x - 6) ----> x^2 + x - 6 = x^2 ----> x - 6 = 0 ----> x = 6-Find the value at x = 0:g(0) = 0Therefore, g(x) passes through (0,0)Find the values (if any) where g(x) = 00 = x² / (x² + x - 6)x² = 0x = 0Therefore, the only zero is (0,0).......

To find the horizontal asymptote use the fact that when x is very large the x^2 on top and bottom dominate the other terms which can be ignored.

Edit. Having got that the horizontal asymptote is y = 1, the curve crosses this when it has the same y value, i.e.

1 = (x^2) / (x^2 + x - 6) ----> x^2 + x - 6 = x^2 ----> x - 6 = 0 ----> x = 6

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Find the value at x = 0:
g(0) = 0
Therefore, g(x) passes through (0,0)

Find the values (if any) where g(x) = 0
0 = x² / (x² + x - 6)
x² = 0
x = 0
Therefore, the only zero is (0,0).

Find the vertical asymptotes:
Set the denominator equal to zero:
x² + x - 6 = 0
(x + 3)(x - 2) = 0
x = -3, 2

Determine the end behavior for the vertical asymptotes:
Left of -3: g(-4) = +/+ = +∞
Right of -3: g(-2) = +/- = -∞

Left of 2: g(1) = +/- = -∞
Right 2: g(3) = +/+ = +∞

Find the horizontal asymptote(s):
lim{x→+∞} g(x) = ∞/∞ = 1
lim{x→-∞} g(x) = ∞/∞ = 1

You should be able to draw a sketch now with that information.

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just find limit x--> +-infinity g(x)and if comes to be finite it has a horizontal asymptote
find y----> +-infinity and x comes finite it has a vertical asymptote

find m = limit x -----> +-infinity [g(x)/x] and it comes to be finite then it may have an inclined asymptote

find c = limit x---> +- infinity [g(x) - m/x] and it comes to be finite then it has an inclined asymptote which is y = mx + c

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