First, a quick reminder on the definition of a root. A root is a value that can be plugged into a function to return zero. For example, with the function y=2-x, the only root would be x = 2.

If you come across the following function, how do you find the roots?

y = [(2x - 3)(x + 3)] / [(x)(x - 2)]

Steps:

1) Set each factor in the numerator AND denominator to equal zero.
2) Solve for x.

Numerator Factors:

2x -3 = 0

2x = 3

x = 3/2

AND

x + 3 = 0

x = -3

NOTE: x = 3/2 and x = -3 become our x-intercepts (roots of the numerator). So, x = 3/2 and x = -3 are two specific locations where the graph will cross the x-axis for the above function.

Denominator Factors:

x = 0

AND

x - 2 = 0

x = 2

NOTE: x = 0 and x = 2 in the denominator become our vertical asymptotes (roots of the denominator). So, there is a vertical asymptote at x = 0 and x = 2 for the above function.

Here's a geometric view of what the above function looks like including BOTH x-intercepts and BOTH vertical asymptotes:



RULE TO KNOW:

After doing away with any common factors from the numerator and denominator, you will be able to locate the vertical asymptotes. REMEMBER: The vertical asymptotes are the roots of the denominator and the x-intercepts are the roots of the numerator.