Graphical Behavior

Holes

Example 2

Find the points of discontinuity for the rational function.

frac1

Step 1. Factor the rational function completely.

This function is in factored form.

Step 2. Determine the points of discontinuity by setting the factors in the denominator equal to zero and solving.

x – 2 = 0 or x + 3 = 0

Unequal or unequal2

Step 3. Simplify the rational function.

cancel
solution

Step 4. Determine the vertical asymptote.

x = -3 is a vertical asymptote because after simplifying the function it is still a zero of the denominator.



Step 5. Determine the coordinates of the hole.

There is a hole at x = 2 because the factor (x - 2) is eliminated from the denominator during simplification. To determine the y-value, plug x = 2 into the simplified function and solve.

solution1

Coordinates of the hole: (2,0)