Graphing Rational Functions

A rational Function is

To graphing a rational function, you must determine if any of the following exist; if they do exist, then identify them.

  1. Intercepts (both x-and y-intercepts)
  2. Asymptotes ( vertical, horizontal, and slant)
  3. Identify any holes if they exist.

To get started:

1st: Factor the numerator and denominator. Reduce if possible.

If the function reduces, then there will be a hole in the graph.

To find the coordinates of the hole: (a) Set the factor(s) that was canceled equal to zero

(b) Substitute this value for “x” in the reduced equation to find the

y-value of the ordered pair.

IF THE FUNCTION REDUCES, THEN USE THE REDUCED FUNCTION TO FIND THE FOLLOWING:

2nd: Find the intercepts if there are any.

To find the x-intercept: Let and solve for. If this statement is false, then there is no x-intercept.

If you get imaginary roots, then the graph does not cross the x-axis.

To find the y-intercept: Let and solve for. If the resulting expression is undefined, then there is no

y-intercept.

3rd: Find the asymptotes:

To find the vertical asymptotes, set the factored denominator and solve.

The resulting equations will be the vertical asymptotes. (Note: Graphs can NEVER cross vertical asymptotes.)

Note: Multiple equal factors in the denominators will define multiple vertical asymptotes. Ex: If the same factor occurs twice, the equation has a double vertical asymptote.

  • If a vertical asymptote occurs an even number of times, then the function will approach the asymptote from both sides and go in the same direction.
  • If a vertical asymptote occurs an odd number of times, then the function will approach the asymptote from both sides and go in opposite directions.

To find the horizontal asymptote, compare the degree of the numerator to the degree of the denominator

  • If the degree of the numerator is less than the degree of the denominator, the HA is the x-axis (y = 0)
  • If the degreeof the numerator is greater than the degree of the denominator, there is no HA. (If this happens, check to see if there is a slant asymptote.)
  • If the degreeof the numerator is equal to the degree of the denominator, then the equation of the HA is

(Note: Graphs CAN cross horizontal asymptotes. To determine if the function crosses the HA, set the

function equal to the HA and solve. If you get a value(s) for x, then it crosses the HA at this x-value. If this

statement is false, then it does not cross the HA)

To find a slant asymptote, divide the numerator by the denominator. The slant asymptote will be

(Ignoring the remainder)

(Note: Graphs CAN cross a slant asymptote. To determine if the function crosses the SA, write the function

equal to the SA and solve. If you get a value(s) for x, then it crosses the SA at this x-value. If this statement is

false then it does not cross the SA)

**** We will not have graphs that cross the slant asymptote in Advanced Precalculus.****

PRECALCULUS ADVANCED

RATIONAL FUNCTIONS

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