# 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: $$\displaystyle \left(4,\frac{1}{20} \right)$$