key term - Hole

Definition

A hole in the context of rational functions refers to a point on the graph where the function is undefined due to the cancellation of common factors in the numerator and denominator. This results in a gap in the graph at that particular x-value, where the function does not have a defined output even though the limit can be approached from both sides. Holes are important because they indicate where discontinuities occur without vertical asymptotes, highlighting a unique characteristic of rational functions.

5 Must Know Facts For Your Next Test

1. Holes occur when both the numerator and denominator of a rational function share a common factor that can be canceled out.
2. To find holes, set the common factor equal to zero to determine the x-value where the hole occurs.
3. The y-value at the hole can be found by substituting the x-value back into the simplified version of the rational function, after canceling the common factors.
4. Holes do not affect the end behavior of rational functions, as they do not create vertical asymptotes.
5. In a graph, holes are represented as open circles indicating that the function does not include those points.

Review Questions

• How do you identify a hole in a rational function's graph, and what steps would you take to find its coordinates?
  • To identify a hole in a rational function, first look for any common factors in both the numerator and denominator that can be canceled. After canceling these factors, set the canceled factor equal to zero to find the x-coordinate of the hole. Then, substitute this x-value into the simplified version of the function to find the corresponding y-coordinate. This results in coordinates for the hole represented as (x, y).
• Discuss how holes differ from vertical asymptotes in terms of their impact on a rational function's graph.
  • Holes differ from vertical asymptotes in that holes indicate points where a function is undefined due to cancellation of factors, while vertical asymptotes represent points where the function approaches infinity. Holes create gaps on the graph with open circles, whereas vertical asymptotes create lines that the graph approaches but never crosses. Therefore, while both indicate discontinuities, holes are removable discontinuities that can be resolved by simplification, whereas vertical asymptotes signal non-removable discontinuities.
• Evaluate how understanding holes and their behavior within rational functions can help in sketching accurate graphs.
  • Understanding holes is crucial for sketching accurate graphs of rational functions since it allows you to identify points where the function is undefined and cannot be included on your graph. By recognizing where these gaps occur, you can accurately depict how the graph behaves around those points and adjust your sketch accordingly. Additionally, being aware of holes helps clarify the overall structure of the function, including its end behavior and any nearby asymptotic behavior, providing a comprehensive understanding that enhances graphing skills.

Related terms

Rational Function:

A function that can be expressed as the ratio of two polynomials.

Vertical Asymptote: A line that the graph approaches but never touches, occurring when the denominator of a rational function is zero and not canceled by a common factor.

Discontinuity: A point at which a function is not continuous, which can be due to holes or asymptotes.