Honors Algebra 2: Extra Practice on Rational Functions -- Asymptotes and Holes in the Graph

• Class: Honors Algebra 2
• Author: Peter Atlas
• Algebra and Trigonometry: Structure and Method, Brown

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For each of the following, identify all vertical asymptotes (if any), horizontal asymptotes (if any), and the coordinates of holes in the graph (if any).

1. \( \displaystyle f(x) = \frac{1}{x-2} \)
   Solution

   V.A.: \(x = 2\); H.A.: \(y = 0\); Holes: none

2. \( \displaystyle f(x) = \frac{3}{2x+1} \)
   Solution

   V.A.: \( \displaystyle x = - \frac{1}{2}\); H.A.: \( \displaystyle y = 0\); Holes: none

3. \( \displaystyle f(x) = \frac{x+3}{x^2 + 4x + 3} \)
   Solution

   V.A.: \( \displaystyle x = - 1\); H.A.: \( \displaystyle y = 0\); Holes: \( \displaystyle \left(-3,-\frac{1}{2} \right)\)

4. \( \displaystyle f(x) = \frac{x^3 - 4x}{x^2 - 4} \)
   Solution

   V.A.: none; H.A.: none; Holes: \( (2, 2)\) and \( (-2, -2)\)

5. \( \displaystyle f(x) = \frac{3x^2 - 8x - 3}{x^2 - x - 6} \)
   Solution

   V.A.: \( \displaystyle x = - 2\); H.A.: \( \displaystyle y = 3\); Holes: \( \displaystyle \left(3,2 \right)\)

6. \( \displaystyle f(x) = \frac{5x^2 + 5ax - bx - ab}{x^2-a^2} \)
   Solution

   V.A.: \( \displaystyle x = a\); H.A.: \( \displaystyle y = 5\); Holes: \( \displaystyle \left(-a,\frac{5a+b}{2a} \right)\)

7. \( \displaystyle f(x) = \frac{4x-16}{x^3 + 6x^2 - 16x - 96} \)
   Solution

   V.A.: \( \displaystyle x = -6\) and \( x = -4 \); H.A.: \( \displaystyle y = 0\); Holes: \( \displaystyle \left(4,\frac{1}{20} \right)\)