upgrade

📈college algebra review

Quadratic-over-Linear

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A quadratic-over-linear function is a rational function where the numerator is a quadratic expression and the denominator is a linear expression. This type of function is important in the study of rational functions as it exhibits unique characteristics and behaviors that are crucial to understand.

5 Must Know Facts For Your Next Test

  1. Quadratic-over-linear functions can exhibit a variety of behaviors, including having asymptotes, points of discontinuity, and local extrema.
  2. The degree of the numerator (quadratic) is higher than the degree of the denominator (linear), which means the function will approach a horizontal asymptote as the input variable approaches positive or negative infinity.
  3. The behavior of a quadratic-over-linear function is heavily influenced by the coefficients and constants in both the numerator and denominator.
  4. Quadratic-over-linear functions can be used to model a wide range of real-world phenomena, such as population growth, radioactive decay, and the motion of objects under the influence of gravity.
  5. Understanding the properties and graph of a quadratic-over-linear function is essential for solving problems involving rational functions and their applications.

Review Questions

  • Explain the key features of a quadratic-over-linear function and how they differ from other types of rational functions.
    • A quadratic-over-linear function is a rational function where the numerator is a quadratic expression and the denominator is a linear expression. This means the function will have a higher degree in the numerator than the denominator, which can lead to unique behaviors such as the presence of asymptotes, points of discontinuity, and local extrema. The specific features of a quadratic-over-linear function, like the location and number of asymptotes and the shape of the graph, are heavily influenced by the coefficients and constants in both the numerator and denominator, setting them apart from other types of rational functions.
  • Describe how the degree difference between the numerator and denominator affects the behavior of a quadratic-over-linear function.
    • The fact that the degree of the numerator (quadratic) is higher than the degree of the denominator (linear) is a key characteristic of a quadratic-over-linear function. This means the function will approach a horizontal asymptote as the input variable approaches positive or negative infinity. The specific location and number of asymptotes, as well as the overall shape of the graph, are determined by the coefficients and constants in both the numerator and denominator. Understanding this degree difference and its implications is crucial for analyzing the properties and behavior of quadratic-over-linear functions.
  • Explain how quadratic-over-linear functions can be used to model real-world phenomena and the importance of this application.
    • Quadratic-over-linear functions can be used to model a wide range of real-world phenomena, such as population growth, radioactive decay, and the motion of objects under the influence of gravity. The ability to accurately represent these types of situations using quadratic-over-linear functions is essential for understanding and predicting the behavior of these systems. By analyzing the specific characteristics of a quadratic-over-linear function, such as its asymptotes, points of discontinuity, and local extrema, we can gain valuable insights into the underlying processes and dynamics of the real-world problem being modeled. This application highlights the importance of thoroughly understanding the properties and graph of a quadratic-over-linear function, as it allows us to make informed decisions and draw meaningful conclusions about the systems we study.