A rational function is a quotient of two polynomials: where and are polynomials of degree n and m...
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A rational function is a quotient of two polynomials: where and are polynomials of degree n and m respectively. Most questions about a rational function can be answered by first factoring the numerator and denominator Example: n m p x f x q x n p x m q x 2 2 4 21 6 x x y x x
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Transcript of A rational function is a quotient of two polynomials: where and are polynomials of degree n and m...
A rational function is a quotient of two polynomials:
where and are polynomials of degree n and m respectively.
Most questions about a rational function can be answered by first factoring the numerator and denominator
Example:
n
m
p xf x
q x
np x mq x
2
2
4 21
6
x xy
x x
It’s all about finding the zeros of the numerator and the denominator.
x (numerator zeros!) & y intercepts (x = 0)
domain/vertical asymptotes (denominator zeros!)
end behavior/horizontal asymptotes/ limits at ∞
2
2
3 74 21
6 2 3
x xx xy
x x x x
More on Asymptotes
For vertical asymptotes check the sign of to the left and right of the asymptote (LHL and RHL limits are useful here!)
For end behavior/horizontal asymptotes, to evaluate the limit first factor out the largest power of x in the denominator and use the result
Asymptotes “shape” the graph of a rational function!
f x
limx
f x
1lim 0x x
End Behavior Examples (3 cases)
1. (n < m)
2. (n = m)
3. (n > m)
22
291
9lim lim ?
3 23 2 1x x
x x xx x
x x
22
2 322 3lim lim ?
2 13 2 1 3x x
x x xx x
x x
2 33lim lim ?
22 1x x
xx xx
x
Putting it all together- Graphing Rational Functions
1. Factor numerator & denominator.2. Find & plot x (numerator zeros) and y intercepts3. Find domain/vertical asymptotes (denominator
zeros) – sketch lines (check signs to left and right of vertical asymptotes – use sign lines)
4. End behavior/horizontal asymptotes: (divide by largest power of x in denominator) - sketch line
5. If needed plot additional points to fill in details
limx
f x
More Examples
1.
2.
3.
2
2
4 21
2
x xy
x x
2
2
2 2
4
xy
x
3 2
3
2 8
1
x x xy
x
Think! How can you factor thesecubic equations?
