Math Homework 2.6 Solution Key

Example 1: Find all horizontal and vertical asymptotes, and holes in the graph. Also, find the zeros of the function and the y-intercept. a. f(x) = 2x² / (x²-1) Horizontal Asymptote (HA): y = 2 Vertical Asymptotes (VA): x = ±1 Holes: None Zeros: (0,0) y-intercept: (0,0) b. f(x) = (x²+x-2) / (x²-x-6) = (x+2)(x-1) / (x+2)(x-3) HA: y = 1 VA: x = 3 Holes: x = -2 (To find the y-coordinate of the hole, plug x = -2 into the reduced rational function) Zeros: (1,0) y-intercept: (0, 1/3) c. f(x) = (x²-x-12) / (x²+4x+3) = (x-4)(x+3) / (x+1)(x+3) HA: y = 1 VA: x = -1 Holes: x = -3 (To find the y-coordinate of the hole, plug x = -3 into the reduced rational function) Zeros: (4,0) y-intercept: (0, -4) Section 2.6 Pg. 152-155 #23abce, 43, 47, 49, 52 Section 2.7 Pg. 161 #9, 15, 17 Example 2: Write a rational function f that has the specified characteristics. There are many correct answers. The only rule here...you can't create any zeros that aren't already given! Vertical Asymptote: x = -1 Horizontal Asymptote: y = 0 Zero: x = 2 f(x) = (x-2) / (x+1)² Example 3: For the function f, find (a) the domain, (b) the vertical asymptote(s) and (c) the horizontal asymptote. f(x) = (3x³+7x+2) / (-4x³+5) Domain: -4x³ + 5 ≠ 0 x³ ≠ 5/4 x ≠ ∛(5/4) (-∞, ∛(5/4)) ∪ (∛(5/4), ∞) VA: x = ∛(5/4) HA: y = -3/4 End Behavior/Horizontal Asymptote: x → ∞, f(x) → -3/4 x → -∞, f(x) → -3/4 Behavior Around Vertical Asymptotes: x → (∛(5/4))⁺, f(x) → ∞ x → (∛(5/4))⁻, f(x) → -∞ Example 4: Find the end behavior and the behavior around the vertical asymptote(s). Graph these on your graphing calculator to assist with these answers! f(x) = (2x+1) / (x+1) x → ∞, f(x) → 2 x → -∞, f(x) → 2 x → -1⁺, f(x) → ∞ x → -1⁻, f(x) → -∞ Section 2.6 Pg. 152-155 #23abce, 43, 47, 49, 52 Section 2.7 Pg. 161 #9, 15, 17 Graphing a Rational Function BY HAND Example 5: Graph f(x) = 1 / (x+6) using intercepts, asymptotes, and end behavior X-Intercepts (Zeros): None Y-Intercept: (0, 1/6) HA: y = 0 VA: x = -6 Holes: None End Behavior: x -> ∞, f(x) -> 0; x -> -∞, f(x) -> 0 Behavior Around VA: x -> -6⁺, f(x) -> ∞; x -> -6⁻, f(x) -> -∞ Example 6: Graph f(x) = x / (x²-9) using intercepts, asymptotes, and end behavior X-Intercepts (Zeros): (0, 0) Y-Intercept: (0, 0) HA: y = 0 VA: x = ±3 Holes: None End Behavior: x -> ∞, f(x) -> 0; x -> -∞, f(x) -> 0 Behavior Around VA: x -> 3⁺, f(x) -> ∞; x -> 3⁻, f(x) -> -∞; x -> -3⁺, f(x) -> -∞; x -> -3⁻, f(x) -> ∞ Section 2.6 Pg. 152-155 #23abce, 43, 47, 49, 52 Section 2.7 Pg. 161 #9, 15, 17