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7.9 – Graphing Rational Functions

Key Terms

  • Sign Chart – A chart that records information about a rational function’s values to help draw its graph.
    • The zeros of the function are marked on the sign chart, and test numbers are used to find whether the function is negative or positive for values of x between the zeros.
  • Singular Point – A point at which the graph of a rational function has a “hole.”
    • If a graph has a singular point at x = b, then the function has a factor of (x – b) in both its numerator and denominator.
  • Zero – The value of x for which a function F(x) equals zero or crosses the x-axis.


Rational Function Graphs
Alg2B 7.9 Graphing ChartAlg2B 7.9 1 ZeroAlg2B 7.9 No Zeros


Steps for Making a Sign Chart
  • Step 1: Find all of the function’s zeros and vertical asymptotes, plot them on a number line, and label them as zeros or asymptotes.
  • Step 2: Choose one x-value on either side of each asymptote and zero.
    • You only need to test one number on either side of each zero or vertical asymptote.
    • Check whether the function is positive or negative at these x-values by substituting them into the function’s equation and solving for (x, y).
      • If the function is positive at a test number, it will remain positive until it reaches a zero or a vertical asymptote.
      • If the function is negative at a test number, it will remain negative until it reaches a zero or an asymptote.
    • You must substitute the SAME value for every x in the function’s equation when testing.
  • Step 3: Use the results of Step 2 to fill in the signs on your sign chart.
    • The sign of a rational function does not always change at every vertical asymptote and zero.
  • Step 4: Make a graph.
    • The graph of a rational function is curved, not a line.
Example 1

  • Step 1. Zeros and Asymptotes
    • In this case, G(x) has a vertical asymptote at x = -1 and a zero at x = 0

Alg2B 7.9 Sign Chart 1

  • Step 2. Choose values on each side
    • x = -2, x = -0.5, and x = 1 can be used as test numbers

Alg2B 7.9 Sign Chart 2


  • Step 3: Fill in Sign Chart

Alg2B 7.9 Sign Chart 3


  • Step 4. Graph the zeros, asymptotes, points, and curves

Alg2B 7.9 Sign Chart Graph


Singular Points
  • A graph with a zero and vertical asymptote in the same place.
  • It creates a hole in the graph.
  • Example

Alg2B 7.9 Singular Point

  • When you factor out the common factor (x + 1), you still need to account for it in your domain!
    • So, x ≠ –2, x ≠ –1, and x  1 
    • The –1 is for the factored-out as factor (x + 1), but it still exists in the domain.


  • Graph the following equation by making a sign chart.
    Alg2B 7.9 ex. Sign Chart 1 Alg2B 7.9 ex. Sign Chart 2 Alg2B 7.9 ex. Sign Chart 3 Alg2B 7.9 ex. Sign Chart 4

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