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# Pre-Cal 30S January 20, 2009

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More on graphing rational functions.

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### Pre-Cal 30S January 20, 2009

1. 1. Lines Approaching Inﬁnity graphing rational functions asintoto [asymptote] by ﬂickr user zinov
2. 2. Graphing Rational Functions Functions of the form where a(x) and b(x) are polynomial functions. Examples
3. 3. Graphing Rational Functions Appearance Where n is odd, the Where n is even, the graph looks like this: graph looks like this:
4. 4. Graphing Rational Functions Sketching (7 steps) Step 1: Find the y-intercept (let x = 0) Step 2: Factor everything. (Use rational roots theorem if necessary.) Step 3: Find the roots of the function by ﬁnding the roots of the numerator a(x). Step 4: Find the vertical asymptotes by ﬁnding the roots of the denominator b(x).
5. 5. Graphing Rational Functions Step 5: Find the horizontal asymptotes by dividing each term in the function by the highest power of x, and take the limit as x goes to inﬁnity. (Use the UNfactored form.) You will ﬁnd that, in general, there are three possible results: i When [degree of numerator < degree of denominator] the horizontal asymptote is y = 0. ii When [degree of numerator = degree of denominator] the H.A. is the ratio leading coefﬁcient of a(x) leading coefﬁcient of b(x) iii When [degree of numerator > degree of denominator] there is no horizontal asymptote; however there may be a slant asymptote or a hole in the graph.
6. 6. Graphing Rational Functions Sketching (7 steps) Step 6: Determine the sign of the function over the intervals deﬁned by the roots and vertical asymptotes. (Use the factored form.) Step 7: Sketch the graph.
7. 7. Graphing Rational Functions Sketching: Example 1 of 4 Step 5: Step 1: Step 6: Step 2: Step 3: Step 7: Step 4:
8. 8. Step 5:
9. 9. Sketch the graph of
10. 10. Graphing Rational Functions Sketching: Example 2 of 4 Step 5: Step 1: Step 6: Step 2: Step 7: Step 3: Step 4:
11. 11. Step 5:
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