Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: ote, the graph of f nevertheless crosses the x-axis at (0, 0). The myth that graphs of rational functions can’t cross their horizontal asymptotes is completely false, as we shall see again in our next example. Example 4.2.2. Sketch a detailed graph of g (x) = 2x2 − 3x − 5 . x2 − x − 6 Solution. 1. Setting x2 − x − 6 = 0 gives x = −2 and x = 3. Our domain is (−∞, −2) ∪ (−2, 3) ∪ (3, ∞). 2. Factoring g (x) gives g (x) = (2x−5)(x+1) (x−3)(x+2) . There is no cancellation, so g (x) is in lowest terms. 3. To find the x-intercept we set y = g (x) = 0. Using the factored form of g (x) above, we find the zeros to be the solutions of (2x − 5)(x + 1) = 0. We obtain x = 5 and x = −1. Since 2 both of these numbers are in the domain of g , we have two x-intercepts, 5 , 0 and (−1, 0). 2 To find the y -intercept, we set x = 0 and find y = g (0) = 5 , so our y -intercept is 0, 5 . 6 6 4. Since g (x) was given to us in lowest terms, we have, once again by Theorem 4.1 vertical x−5)(x+1) asymptotes x = −2 and x = 3....
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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