Chapter 2 section 6 and 7 Rational Functions and Asymptotes, Graphing Rational Functions Rational Functions N(x) and D(x) are poly. ( ) ( ) ( ) Domain: all real numbers except x-values that make the denominator zero Look at all examples in section 6 and 7 Vertical Asymptotes and Holes For ( ) ( ) ( ) where N(x) and D(x) have no common factors (reduce first) The graph has vertical asymptotes at the zeros of D(x) The graph has a hole at any reduced factor Try this: Example: Find the vertical asymptotes and holes: ( ) answer: vertical asymptote at x = 1, hole (discontinued point) at (-1, -1/2) work: factor and reduce: ( )( ) factor in the denominator (x – 1) is the vertical asymptote: x = 1 factor that reduced out (x + 1) is the hole at x = -1 substitute -1 into reduced equation, , to find the exact point of the hole , therefore the hole is at (-1, -1/2) Horizontal Asymptotes (memorize these rule) For ( ) ( ) 1. If n < m, then y = 0 (Ex. ) 2. If n = m, then (Ex. , then y = ) 3. If n > m, then no horizontal asymptote (Ex. ) Try this: Example Find the domain and asymptotes. ( ) answer: Domain: Vertical Asy: ; Horizontal Asy: ; No holes work: Domain comes from 4x – 6 ≠ 0 Horizontal Asy: , x both to 1st degree therefore coef. is HA Vertical Asy: 4x – 6 = 0 Holes: nothing reduced out Slant Asymptotes (also called, Oblique) (See page 202) If the degree of N(x) is 1 more than D(x), then slant asymptote Use long division to find. This happens when there is no horizontal asy. Steps to graph rational functions (See page 199 box) ( ) ( ) ( ) N(x) and D(x) have no common factors 1. Find y-intercepts, f(0) 2. Find zeros, N(x) = 0 3. Find vertical asymptotes and holes D(x) = 0 4. Find horizontal asymptotes 5. Find slant (oblique) asymptotes 6. Graph points on each side of vertical asy. Example – Find domain, all asymptotes, holes and intercepts and graph with additional points ( ) answer: Domain: (bottom) HA: y = 1 (leading terms) VA: x = 3, x = -3 (bottom) Hole: none (bottom) Slant asy: none x –intercept: (0, 0) (numerator = 0, x2 = 0) y –intercept: (0, 0) (let x = 0) additional points: (-4, 16/7), (-5, 25/16), (-1, -1/8), (1, -1/8) (4, 16/7), (5, 25/16) Example Find Domain, all asymptotes, holes and intercepts and graph with additional points ( ) answer: Reduce: Domain: x ≠ -3, x ≠ 2 HA: y = 0 VA: x = 2 Hole: (-3, -1/5) Slant Asy: none x-intercept: none y-intercept (0, -1/2) additional points: (3, 1) (4, ½) (1, -1) Example Graph ( ) answer: Domain: x ≠ 1 HA: none VA: x = 1 Hole: none Slant Asy.: y = x (long division, disregard remainder) x-int. (2, 0) ( -1, 0) y-int (0, 2) additional points: (3, 2) (3/2, -5/2) (-2, -4/3)
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