# 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.

## Notes

Rational Function Graphs

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

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

• Step 3: Fill in Sign Chart

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

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

• 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.

## Examples

 Graph the following equation by making a sign chart.