📚 Topic Summary
Rational functions are functions that can be written as a ratio of two polynomials. Graphing them involves finding key features like asymptotes (vertical, horizontal, and oblique), intercepts, and holes. Understanding these features allows you to accurately sketch the graph of the function.
This worksheet provides a structured way to practice identifying these features and using them to graph rational functions. By working through these problems, you'll gain confidence in your ability to analyze and graph rational functions effectively.
🧠 Part A: Vocabulary
Match the term with its definition. Write the number of the definition in the blank next to the term.
- Vertical Asymptote: ______
- Horizontal Asymptote: ______
- Oblique Asymptote: ______
- Intercept: ______
- Hole: ______
Definitions:
- The point where a graph crosses the x-axis or y-axis.
- A removable discontinuity in a rational function, occurring when a factor cancels from both the numerator and denominator.
- A line that the graph of a function approaches as $x$ approaches positive or negative infinity, when the degree of the numerator is exactly one more than the degree of the denominator.
- A line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$.
- A line $y = b$ that the graph of a function approaches as $x$ approaches positive or negative infinity.
✏️ Part B: Fill in the Blanks
Complete the following sentences:
When graphing rational functions, it's important to first identify the ________. These occur where the denominator of the rational function equals _______. Next, find the _________, which describes the function's behavior as x approaches positive or negative infinity. A _______ exists when a factor is present in both the numerator and denominator and cancels out. Finally, identify the _______ by setting x and y to zero, respectively.
🤔 Part C: Critical Thinking
Explain in your own words how the degrees of the numerator and denominator of a rational function determine the existence and location of horizontal or oblique asymptotes.