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Jan. 31, 2023•0 likes•2 views

Jan. 31, 2023•0 likes•2 views

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Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your result. (If an answer does not exist, enter DNE.) Vertical asymptote x=_________ Horizontal asymptote y=__________ slant asymptote y=___________ hole (x, y) = (___, ___) Solution Vertical Asymptotes: 1. Factor the numerator and denominator and reduce the rational expression (x - 2) (x + 2) (2 x + 1) ------------------------------ (x - 1) (x - 2) (x + 2) (2 x + 1) ------------------------------ (x - 1) The vertical asymptote is a vertical line with the equation x = 1. Hole is at (2, f(2)) by virtue of dividing our the x-2 factor A graph can have a (one) slant asymptote or a (one) horizontal asymptote. Since the degree of the numerator is exactly 1 greater than the degree of the denominator, the rational expression has a slant asymptote found by dividing the numerator by the denominator The slant asymptote is y = 2x + 7 .

- 1. Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your result. (If an answer does not exist, enter DNE.) Vertical asymptote x=_________ Horizontal asymptote y=__________ slant asymptote y=___________ hole (x, y) = (___, ___) Solution Vertical Asymptotes: 1. Factor the numerator and denominator and reduce the rational expression (x - 2) (x + 2) (2 x + 1) ------------------------------ (x - 1) (x - 2) (x + 2) (2 x + 1) ------------------------------ (x - 1) The vertical asymptote is a vertical line with the equation x = 1. Hole is at (2, f(2)) by virtue of dividing our the x-2 factor
- 2. A graph can have a (one) slant asymptote or a (one) horizontal asymptote. Since the degree of the numerator is exactly 1 greater than the degree of the denominator, the rational expression has a slant asymptote found by dividing the numerator by the denominator The slant asymptote is y = 2x + 7