Final answer:
To find a hole in the graph of a rational function algebraically, factor the function, identify common factors, determine their zeros, and plug these into the simplified function to find the hole's coordinates.
Step-by-step explanation:
To find a hole of a graph algebraically, you first need to look at the rational function and identify any common factors in the numerator and denominator. A hole occurs at a point where a factor cancels out in the fraction, which translates graphically to a point that the graph doesn't touch, even though it might pass close by.
Here is a step-by-step explanation to find the hole algebraically:
- Factor the numerator and denominator of the function completely.
- Identify any common factors that appear both in the numerator and the denominator.
- The values that cause these common factors to be zero are where the holes will occur.
- To find the exact coordinates of the hole, set the common factor equal to zero to find the x-coordinate, then plug that x-coordinate into the simplified function (without the cancelled factor) to find the corresponding y-coordinate.
This process allows you to find the coordinates of the hole, if there is one, in the graph of a rational function.