📚 Topic Summary
In Algebra 2, understanding rational functions is key. Vertical asymptotes occur where the denominator of a rational function equals zero, but the factor doesn't cancel out with the numerator. These are vertical lines that the function approaches but never crosses. Holes, on the other hand, happen when a factor in the denominator *does* cancel out with a factor in the numerator. This creates a point of discontinuity, or a 'hole', in the graph at that x-value.
Essentially, vertical asymptotes represent values that make the function undefined, causing it to shoot off to infinity (or negative infinity). Holes are removable discontinuities, meaning we can 'fill' them in by simplifying the function.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Vertical Asymptote |
A. A point where the function is undefined due to a common factor in the numerator and denominator. |
| 2. Hole |
B. A line that the function approaches but never crosses, occurring when the denominator equals zero and the factor doesn't cancel. |
| 3. Rational Function |
C. A function that can be written as a ratio of two polynomials. |
| 4. Discontinuity |
D. A point on a function where the graph is not continuous, such as a hole or vertical asymptote. |
| 5. Removable Discontinuity |
E. A discontinuity that can be 'removed' by simplifying the function, creating a continuous graph. |
✏️ Part B: Fill in the Blanks
A __________ occurs when the denominator of a rational function equals zero, and this factor does not cancel with the numerator. A __________ is a point where the function is undefined because a factor in both the numerator and denominator cancels out. This type of discontinuity is called __________, because the function can be simplified to 'fill' the gap.
🤔 Part C: Critical Thinking
Explain, in your own words, the difference between a vertical asymptote and a hole in a rational function. Provide an example function to support your explanation.