Limits D6 Graphs of Rational Functions Solutions

Honors Pre-Calculus Limits D6 Worksheet Name: Examining Graphs of Rational Functions Graph the following rational function to find the domain, range, intercepts, asymptotes, holes in the graph, type of discontinuity and where it occurs, end behavior, and behavior around vertical asymptotes. 1. f(x) = x / (x² - 9) = x / ((x + 3)(x - 3)) Domain: (-∞, -3) U (-3, 3) U (3, ∞) Range: (-∞, 0) U (0, ∞) y-intercept: (0, 0) x-intercept(s): (0, 0) Horizontal Asymptote: y = 0 Slant Asymptote: none Vertical Asymptote(s): x = ±3 Hole(s) in the graph: none Discontinuity: Line at x = ±3 End Behavior: As x → -∞, f(x) → 0 As x → ∞, f(x) → 0 Behavior around V.A.: As x → -3⁻, f(x) → ∞ As x → -3⁺, f(x) → -∞ As x → 3⁻, f(x) → -∞ As x → 3⁺, f(x) → ∞ 2. f(x) = (2x² - 5x + 2) / (2x² - x - 6) = (2x - 1)(x - 2) / ((2x + 3)(x - 2)) Domain: (-∞, -3/2) U (-3/2, 2) U (2, ∞) Range: (-∞, 3/7] U (3/7, 1) U (1, ∞) y-intercept: (0, -1/3) x-intercept(s): (1/2, 0) Horizontal Asymptote: y = 1 Slant Asymptote: none Vertical Asymptote(s): x = -3/2 Hole(s) in the graph: (2, 3/7) Discontinuity: Line at x = -3/2, Point at x = 2 End Behavior: As x → -∞, f(x) → 1 As x → ∞, f(x) → 1 Behavior around V.A.: As x → -3/2⁻, f(x) → ∞ As x → -3/2⁺, f(x) → -∞ Examining Graphs of Rational Functions Graph the following rational function to find the domain, range, intercepts, asymptotes, holes in the graph, type of discontinuity and where it occurs, end behavior, and behavior around vertical asymptotes. 3. f(x) = (5x + 20) / (x² + x - 12) = 5(x + 4) / ((x + 4)(x - 3)) Domain: (-∞, -4) U (-4, 3) U (3, ∞) Range: (-∞, -5/7] U (-5/7, 0) U (0, ∞) y-intercept: (0, -5/3) x-intercept(s): None Horizontal Asymptote: y = 0 Slant Asymptote: None Vertical Asymptote(s): x = 3 Hole(s) in the graph: (-4, -5/7) Discontinuity: Line at x = 3 Point at x = -4 End Behavior: As x → -∞, f(x) → 0 As x → ∞, f(x) → 0 Behavior around V.A.: As x → 3⁻, f(x) → -∞ As x → 3⁺, f(x) → ∞ 4. two graphs f(x) = (2x³ - x² - 2x + 1) / (x² + 3x + 2) = (x² - 1)(2x - 1) / ((x + 1)(x + 2)) Domain: (-∞, -2) U (-2, -1) U (-1, ∞) Range: (-∞, 21.954] U (-0.455, ∞) y-intercept: (0, 1/2) x-intercept(s): (1, 0), (-1/2, 0) Horizontal Asymptote: None Slant Asymptote: y = 2x - 7 Vertical Asymptote(s): x = -2 Hole(s) in the graph: (-1, 6) Discontinuity: Line at x = -2 Point at x = -1 End Behavior: As x → -∞, f(x) → -∞ As x → ∞, f(x) → ∞ Behavior around V.A.: As x → -2⁻, f(x) → ∞ As x → -2⁺, f(x) → -∞ Slant: (x - 1)(2x - 1) / (x + 2) = (2x² - 3x + 1) / (x + 2) Hole: ((-1 - 1)(2(-1) - 1)) / ((-1 + 2)) = -6/1 = -6