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 â€“ 2 = 0 or x + 3 = 0

Step 3. Simplify the rational function.

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.

Coordinates of the hole: (2,0)