Rational Functions and Vertical Asymptotes - AP Precalculus

1. What condition creates a vertical asymptote in a rational function?

   A vertical asymptote occurs where $Q(x) = 0$ and $P(x) \neq 0$. Denominator zero with nonzero numerator creates infinite discontinuity.

2. Identify the vertical asymptote of $f(x) = \frac{1}{x - 3}$.

   Vertical asymptote at $x = 3$. Set denominator $x - 3 = 0$ to find where function is undefined.

3. Determine the vertical asymptote for $f(x) = \frac{x}{x^2 + 4x + 4}$.

   Vertical asymptote at $x = -2$. Factor denominator as $(x+2)^2$; repeated zero at $x=-2$.

4. State the vertical asymptote of $f(x) = \frac{5}{x^2 - 6x + 9}$.

   Vertical asymptote at $x = 3$. Factor denominator as $(x-3)^2$; repeated zero at $x=3$.

5. Identify the vertical asymptote for $f(x) = \frac{x^2 - 9}{x - 3}$.

   Removable discontinuity at $x = 3$. Common factor $ (x-3) $ cancels, creating a hole at $x=3$.

6. Find the vertical asymptote for $f(x) = \frac{1}{x^2 + x}$.

   Vertical asymptotes at $x = 0$ and $x = -1$. Factor denominator as $x(x+1)$ and set each factor to zero.

7. Find the vertical asymptote for $f(x) = \frac{2x}{x^2 - 9}$.

   Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.

8. Determine the vertical asymptote for $f(x) = \frac{x^2}{x^2 - 1}$.

   Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator as $(x-1)(x+1)$ and solve for zeros.

9. What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 25}$?

   Vertical asymptotes at $x = 5$ and $x = -5$. Factor denominator as $(x-5)(x+5)$ and solve for zeros.

10. What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 9}$?

    Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.

11. Does $f(x) = \frac{1}{x^2 + 4}$ have a vertical asymptote?

    No vertical asymptotes. Denominator $x^2 + 4$ has no real zeros since discriminant is negative.

12. Identify the vertical asymptote for $f(x) = \frac{2x - 1}{x^2 - 4x}$.

    Vertical asymptotes at $x = 0$ and $x = 4$. Factor denominator as $x(x-4)$ and set each factor to zero.

13. Does $f(x) = \frac{x + 2}{x^2 + 1}$ have a vertical asymptote?

    No vertical asymptotes. Denominator $x^2 + 1$ has no real zeros since it's always positive.

14. What is the vertical asymptote of $f(x) = \frac{1}{x^2 + 2x + 1}$?

    Vertical asymptote at $x = -1$. Factor denominator as $(x+1)^2$; repeated zero at $x=-1$.

15. Find the vertical asymptote for $f(x) = \frac{1}{x^2 - 6x + 9}$.

    Vertical asymptote at $x = 3$. Factor denominator as $(x-3)^2$; repeated zero at $x=3$.

16. What is the vertical asymptote of $f(x) = \frac{x + 1}{x^2 - x - 6}$?

    Vertical asymptotes at $x = 3$ and $x = -2$. Factor denominator $x^2 - x - 6 = (x-3)(x+2)$ and solve.

17. State the vertical asymptote of $f(x) = \frac{x^2}{x^2 - 4}$.

    Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and set each factor to zero.

18. Identify the vertical asymptote for $f(x) = \frac{x - 4}{x^3 - 8}$.

    Vertical asymptote at $x = 2$. Factor $x^3 - 8 = (x-2)(x^2+2x+4)$; only real zero is $x=2$.

19. State the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 + x - 2}$.

    Vertical asymptotes at $x = 1$ and $x = -2$. Factor denominator as $(x-1)(x+2)$ and solve for zeros.

20. Find the vertical asymptote for $f(x) = \frac{2x^2}{x^2 - 9}$.

    Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and solve for zeros.

21. State the vertical asymptote of $f(x) = \frac{x}{x^2 - 16}$.

    Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator as $(x-4)(x+4)$ and solve for zeros.

22. What is the vertical asymptote of $f(x) = \frac{x}{x^3 - 27}$?

    Vertical asymptote at $x = 3$. Factor $x^3 - 27 = (x-3)(x^2+3x+9)$; only real zero is $x=3$.

23. Determine the vertical asymptote for $f(x) = \frac{x + 2}{x^2 - 2x - 3}$.

    Vertical asymptotes at $x = 3$ and $x = -1$. Factor denominator as $(x-3)(x+1)$ and solve for zeros.

24. Find the vertical asymptote for $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$.

    Vertical asymptotes at $x = 2$ and $x = 3$. Factor denominator as $(x-2)(x-3)$ and solve for zeros.

25. What is the definition of a rational function?

    A function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials. Two polynomials create a fraction with specific asymptotic behavior.

26. State the vertical asymptote for $f(x) = \frac{x^2 - 1}{x - 1}$.

    Removable discontinuity at $x = 1$. Common factor $(x-1)$ cancels, leaving a hole instead of asymptote.

27. Determine the vertical asymptotes for $f(x) = \frac{1}{x^2 - 1}$.

    Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator as $(x-1)(x+1)$ and set each factor to zero.

28. What is the vertical asymptote of $f(x) = \frac{2}{x^2 - 4}$?

    Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and solve for zeros.

29. State the vertical asymptote of $f(x) = \frac{3}{x^2 - 9x}$.

    Vertical asymptotes at $x = 0$ and $x = 9$. Factor denominator as $x(x-9)$ and set each factor to zero.

30. What is the vertical asymptote of $f(x) = \frac{2x}{x^2 - 1}$?

    Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator as $(x-1)(x+1)$ and solve for zeros.

31. Determine the vertical asymptote for $f(x) = \frac{3x}{x^2 - 16}$.

    Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator as $(x-4)(x+4)$ and solve for zeros.

32. Identify the vertical asymptote for $f(x) = \frac{x}{x^2 - 2x}$.

    Vertical asymptotes at $x = 0$ and $x = 2$. Factor denominator as $x(x-2)$ and set each factor to zero.

33. Find the vertical asymptote for $f(x) = \frac{4}{x^2 - 4x + 3}$.

    Vertical asymptotes at $x = 1$ and $x = 3$. Factor denominator as $(x-1)(x-3)$ and solve for zeros.

34. What is the vertical asymptote of $f(x) = \frac{5}{x^2 + 2x}$?

    Vertical asymptotes at $x = 0$ and $x = -2$. Factor denominator as $x(x+2)$ and solve for zeros.

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