To sketch the graph of a function k(x)=\frac{f(x)}{g(x)}:

  1. Find the intercepts:
    1. X-intercepts, set y=0 (there can be multiple)
    2. Y-intercept, set x=0 (there can be only one)
  2. Factorise the numerator and denominator if possible:
    1. Sign table: determine where the function is negative and where it is positive
  3. Find the Vertical asymptotes:
    1. This occur if the function in the denominator is equal to zero, i.e g(x) = 0, AND that in the numerator must not be zero, i.e f(x)\ne 0.
  4. Find any Horizontal asymptote:
    1. If the degree of the function in the numerator, i.e f(x), is less than the degree of the function in the denominator, i.e g(x), then the horizontal asymptote is the line y = 0.
    2. If the degree of the function in the numerator, i.e f(x), is equal to the degree of the function in the denominator, i.e g(x), say for example, the degree of f(x) and g(x) is n for some non-negative n element of integers, then there is a horizontal asymptote. This will still be proven by long division, this is just for you to be able to make reasonable assumptions on how your function will behave or look like.
  5. Find and slant asymptote:
    1. If the degree of the function in the numerator exceeds that in the denominator by one, that is,
      \text{Degree of }f(x) = 1 + \text{Degree of }g(x).
  6. Find and quadratic asymptote:
    1. If the degree of the function in the numerator exceeds that in the denominator by two, that is,
      \text{Degree of }f(x) = 2 + \text{Degree of }g(x)
  7. Perform long division
    1. Write \frac{f(x)}{g(x)}=d(x)+\frac{r(x)}{g(x)}).
    2. Find the equations for the slant, horizontal, quadratic and/or vertical asymptotes (Now after noting that you have some type of an asymptote, long division allows you now to rigorously find the equation of that asymptote). The equation of the x\rightarrow \pm \infty asymptote will be equal to d(x).
    3. Find the various limits (Here we are not interested in the concept of a limit in the more technical sense that you will see later, we are just looking at how the graph behaves as x approaches a certain number or as it approaches \pm \infty).
    4. Find the points where the graph crosses the asymptotes by finding when r(x)=0.
  8. Find any point discontinuities:
    1. These occur when the function in the numerator, f(x) and that in the denominator g(x) are BOTH equal to zero, that is, f(x) = g(x) = 0 for some x. These single points are removed from the graph and are replaced with an open circle.
  9. To sketch the graph:
    1. Start by using dashed lines to draw all the asymptotes.
    2. Use the information in the sign table to determine whether the graph is going to + or –  \infty for the vertical asymptotes.
    3. Draw on any x and y intercepts as well as points where the graph crosses the asymptotes.
    4. Draw in any point discontinuities.
    5. Knowing that the graph doesn’t cross any of the axes or asymptotes other than those which you have already marked on, join together all the information you have to sketch the graph
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