How to Graph Rational Functions From Equations in 7 Easy Steps

1. FACTOR the numerator and denominator

  • Or have your graphing calculator do it for you

2. See if there are any HOLES

  • If a term like (x-3) appears in both the numerator and denominator, then cancel them out and note that there is a hole at x=3
  • To find the y-coordinate of the hole, plug it into the simplified equation

3. Find VERTICAL ASYMPTOTES by finding where factors in the denominator equal zero

  • ex: 1/(x+2) has a vertical asymptote at x = -2

4. See if the fraction is TOP HEAVY, BOTTOM HEAVY, OR BALANCED for Non-Vertical (Horizontal and Oblique/Slant) Asymptotes

  • top-heavy (by one degree, like x? / x) = oblique asymptote
  • To find the equation of the oblique asymptote, use long division (ignore the remainder)
  • bottom-heavy = horizontal asymptote at y=0
  • The rational function will just get infinitely smaller
  • balanced = compare leading coefficients for the horizontal asymptote
  • ex: 3(x-1) / 2(x+2)(x+1) has a horizontal asymptote at y = 3/2

5. Find the x-intercepts where the numerator is equal to zero

  • If the numerator is 0, then that means the y-value is 0, which means it is an x-intercept
  • You can also find y-intercepts by plugging in x=0

6. See if the graph passes through any of the non-vertical asymptotes

  • Set the rational function equal to the horizontal or oblique/slant asymptote
  • ex: For (x-1) / (x+2), you would set (x-1) / (x+2) = 0
  • If there are no solutions, then it does not cross the non-vertical asymptote

7. Test each region between the x-intercepts and vertical asymptotes to see if the graph is positive or negative

  • Once you do this, you just fill in the curves, connecting your points and making them hug the asymptotes

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