Graph Rational Functions and Identify Asymptotes - Algebra 2

1. Identify the simplified function for $f(x)=\frac{x^2-1}{x-1}$ for $x\neq 1$.

   $f(x)=x+1$ for $x\neq 1$. After canceling $(x-1)$: $\frac{x^2-1}{x-1}=x+1$.

2. What is the hole (removable discontinuity) rule for a rational function in factored form?

   A common factor cancels; the hole is at that factor's zero. Common factors create holes at their zeros.

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

   $y=x+2$. Factor and cancel: $\frac{x^2-4}{x-2}=x+2$ after division.

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

   $x=2$. Factor denominator: $(x-2)^2$ gives double root.

5. Identify the domain of $f(x)=\frac{1}{(x-2)(x+5)}$.

   All real $x$ except $x=2$ and $x=-5$. Exclude zeros of denominator factors.

6. What is the $x$-intercept rule for a rational function in factored form?

   Zeros of the numerator that are not canceled by the denominator. Numerator zeros give x-intercepts unless cancelled out.

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

   $x=1$. Denominator is zero when $x-1=0$.

8. Find the hole location for $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

   Hole at $x=4$. Common factor $(x-4)$ creates hole.

9. What is the multiplicity effect near a vertical asymptote $(x-a)^k$ in the denominator?

   Odd $k$: opposite infinities; even $k$: same infinity on both sides. Multiplicity affects sign behavior near asymptotes.

10. Identify the horizontal asymptote of $f(x)=\frac{x^2}{x^2-9}$.

    $y=1$. Equal degrees: $\frac{1}{1}=1$.

11. What type of asymptote occurs when $\deg(p)=\deg(q)+1$ for $f(x)=\frac{p(x)}{q(x)}$?

    A slant (oblique) asymptote from polynomial division. Numerator one degree higher creates slant asymptote.

12. Identify the horizontal asymptote of $f(x)=\frac{x+2}{x-1}$.

    $y=1$. Equal degrees: $\frac{1}{1}=1$.

13. What is the $y$-intercept of $f(x)$, if it exists?

    $f(0)$, provided $q(0)\neq 0$. Substitute $x=0$ if denominator is non-zero.

14. What is the general form of a rational function $f(x)$?

    $f(x)=\frac{p(x)}{q(x)}$ where $p(x),q(x)$ are polynomials and $q(x)\neq 0$. Ratio of polynomials with non-zero denominator.

15. Identify the $x$-intercept(s) of $f(x)=\frac{x^2}{x^2-9}$.

    $x=0$. Numerator $x^2$ is zero when $x=0$.

16. What is the multiplicity effect on $x$-intercepts for a factor $(x-a)^k$ in the numerator?

    Odd $k$: crosses; even $k$: touches and turns at $x=a$. Multiplicity determines crossing vs touching behavior.

17. What is the horizontal asymptote when $\deg(p)=\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?

    $y=\frac{\text{leading coeff of }p}{\text{leading coeff of }q}$. Equal degrees give ratio of leading coefficients.

18. Identify the end behavior of $f(x)=\frac{-2x^5+1}{x^4-7}$ as $x\to\pm\infty$.

    No horizontal asymptote; $f(x)\sim -2x$. Numerator degree exceeds denominator by 1.

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

    $x=3$. Denominator is zero when $x-3=0$.

20. What is the key difference between a hole and a vertical asymptote at $x=a$?

    Hole: factor cancels; VA: factor remains in denominator. Cancellation creates holes; non-cancellation creates VAs.

21. What is the end behavior of $f(x)=\frac{3x^2+1}{x^3-5}$ as $x\to\pm\infty$?

    $f(x)\to 0$. Numerator degree less than denominator degree.

22. Identify the vertical asymptote(s) of $f(x)=\frac{x^2}{x^2-9}$.

    $x=-3$ and $x=3$. Factor denominator: $x^2-9=(x-3)(x+3)$.

23. Identify the horizontal asymptote of $f(x)=\frac{x^2-9}{x^2-4x+4}$.

    $y=1$. Equal degrees: $\frac{1}{1}=1$.

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

    $y=x$. Polynomial division: $\frac{x^2+1}{x}=x+\frac{1}{x}$.

25. Identify the $x$-intercept(s) of $f(x)=\frac{x^2-9}{x^2-4x+4}$.

    $x=-3$ and $x=3$. Factor numerator: $(x-3)(x+3)$ gives zeros.

26. Identify the vertical asymptote(s) of $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

    $x=1$. After canceling $(x-4)$, only $x-1=0$ remains.

27. Identify whether $f(x)=\frac{x^2-1}{x-1}$ has a hole or vertical asymptote at $x=1$.

    A hole at $x=1$. Factor $(x-1)$ cancels from numerator and denominator.

28. Identify the $x$-intercept(s) of $f(x)=\frac{(x-2)(x+1)}{x-5}$.

    $x=2$ and $x=-1$. Set numerator factors equal to zero.

29. What is the domain rule for $f(x)=\frac{p(x)}{q(x)}$ before simplifying?

    All real $x$ except where $q(x)=0$. Exclude values where denominator equals zero.

30. Identify whether $f(x)=\frac{x-1}{x-1}$ has a vertical asymptote or a hole at $x=1$.

    A hole at $x=1$. Common factor $(x-1)$ cancels completely.

31. Identify the behavior near $x=2$ for $f(x)=\frac{1}{(x-2)^2}$.

    As $x\to 2^{\pm}$, $f(x)\to +\infty$. Even multiplicity makes both sides approach same infinity.

32. What is the end behavior of $f(x)=\frac{4x^3-x}{2x^3+7}$ as $x\to\pm\infty$?

    $f(x)\to 2$. Equal degrees: $\frac{4}{2}=2$.

33. What is the horizontal asymptote when $\deg(p)<\deg(q)$ for $f(x)=\frac{p(x)}{q(x)}$?

    $y=0$. Lower degree numerator approaches zero at infinity.

34. What is the end behavior idea for a rational function $f(x)$?

    As $x\to\pm\infty$, $f(x)$ approaches its asymptote (if any). Function approaches asymptote as x approaches infinity.

35. Identify whether the graph crosses or touches the $x$-axis at $x=-1$ for $f(x)=\frac{(x+1)^2}{x-3}$.

    Touches (even multiplicity at $x=-1$). Even multiplicity at x-intercept means touching.

36. Find the hole point for $f(x)=\frac{x^2-1}{x-1}$.

    Hole at $\left(1,2\right)$. Substitute $x=1$ into simplified form $x+1$.

37. Identify the horizontal asymptote of $f(x)=\frac{5}{x^2+1}$.

    $y=0$. Degree of numerator less than denominator.

38. Identify the behavior near $x=2$ for $f(x)=\frac{1}{x-2}$.

    As $x\to 2^{-}$, $f(x)\to -\infty$; as $x\to 2^{+}$, $f(x)\to +\infty$. Odd multiplicity creates opposite infinities on each side.

39. Identify the $y$-intercept of $f(x)=\frac{x+2}{x-1}$.

    $f(0)=-2$, so $\left(0,-2\right)$. Substitute $x=0$: $\frac{0+2}{0-1}=-2$.

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

    $x=3$. Denominator $(x-3)^3$ has odd multiplicity 3.

41. What is the vertical asymptote rule for $f(x)=\frac{p(x)}{q(x)}$ in factored form?

    Zeros of $q(x)$ that do not cancel with $p(x)$. Denominator zeros cause vertical asymptotes unless cancelled.

42. What is the correct order of steps to graph $f(x)=\frac{p(x)}{q(x)}$ from a factored form?

    Find holes, VAs, HAs/slant, intercepts, then sketch behavior. Systematic approach ensures all features are identified.

43. Identify the $x$-intercept(s) of $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

    $x=-2$. From remaining numerator factor $(x+2)$.

44. Identify the $x$-intercept of $f(x)=\frac{x+2}{x-1}$.

    $x=-2$. Set numerator equal to zero: $x+2=0$.

45. What is the sign-based test to determine the side behavior near a vertical asymptote $x=a$?

    Check the sign of $f(x)$ on each side using factor signs. Test sign changes across vertical asymptotes.

46. Find the hole point for $f(x)=\frac{(x-4)(x+2)}{(x-4)(x-1)}$.

    Hole at $\left(4,\frac{6}{3}\right)=\left(4,2\right)$. Substitute $x=4$ into simplified function.

47. Identify the horizontal asymptote of $f(x)=\frac{2x^3}{-4x^3+7}$.

    $y=-\frac{1}{2}$. Leading coefficient ratio: $\frac{2}{-4}=-\frac{1}{2}$.

48. Identify the vertical asymptote of $f(x)=\frac{x^2+1}{x}$.

    $x=0$. Denominator is zero when $x=0$.

49. Identify the horizontal asymptote of $f(x)=\frac{(x+1)^2}{(x-3)^3}$.

    $y=0$. Numerator degree less than denominator degree.

50. Identify the vertical asymptote(s) of $f(x)=\frac{(x-2)(x+1)}{(x-5)(x+3)}$.

    $x=5$ and $x=-3$. Set denominator factors equal to zero.

51. Identify whether $f(x)=\frac{x^2-4}{x-2}$ has a hole or vertical asymptote at $x=2$.

    A hole at $x=2$. Factor $(x-2)$ cancels from numerator $(x^2-4)$.

52. Find the hole point for $f(x)=\frac{x^2-4}{x-2}$.

    Hole at $\left(2,4\right)$. Substitute $x=2$ into simplified form $x+2$.

53. What is the horizontal asymptote of $f(x)=\frac{-7x^4+1}{2x^4-3x}$?

    $y=-\frac{7}{2}$. Equal degrees: $\frac{-7}{2}=-\frac{7}{2}$.

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