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# Graphing Rational Functions

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### Graphing Rational Functions

1. 1. January 19, 2010 Graphing Rational Functions Functions of the form where a(x) and b(x) are polynomial functions. Examples
2. 2. January 19, 2010 Graphing Rational Functions Appearance Where n is even, the Where n is odd, the graph looks like this: graph looks like this:
3. 3. January 19, 2010 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).
4. 4. January 19, 2010 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.
5. 5. January 19, 2010 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.
6. 6. January 19, 2010 Graphing Rational Functions Sketching: Example 1 of 4 Step 1: Step 5: Step 6: Step 2: Step 3: Step 7: Step 4:
7. 7. January 19, 2010 Step 5:
8. 8. January 19, 2010 Sketch the graph of
9. 9. January 19, 2010
10. 10. January 19, 2010
11. 11. January 19, 2010 Graphing Rational Functions Sketching: Example 2 of 4 Step 5: Step 1: Step 6: Step 2: Step 7: Step 3: Step 4:
12. 12. January 19, 2010 Step 5: