Linear-Over-Linear

Definition

Linear-over-linear is a type of rational function where the numerator and denominator are both linear functions. This means the numerator and denominator are both first-degree polynomials, with variables raised to the power of 1. The structure of a linear-over-linear rational function allows for unique properties and behaviors that are important to understand in the context of rational functions.

5 Must Know Facts For Your Next Test

1. The graph of a linear-over-linear rational function is a hyperbola, which has a horizontal and vertical asymptote.
2. The domain of a linear-over-linear rational function is all real numbers except the values of x that make the denominator equal to 0.
3. Linear-over-linear rational functions can be used to model a variety of real-world situations, such as inverse variation and rates of change.
4. Transformations of linear-over-linear rational functions, such as shifts, reflections, and scalings, can be used to analyze and sketch their graphs.
5. The end behavior of a linear-over-linear rational function is determined by the relative degrees of the numerator and denominator polynomials.

Review Questions

• Explain the key features of the graph of a linear-over-linear rational function.
  • The graph of a linear-over-linear rational function is a hyperbola, which has a horizontal and vertical asymptote. The horizontal asymptote is determined by the ratio of the constant terms in the numerator and denominator, while the vertical asymptote is determined by the values of x that make the denominator equal to 0. The graph also has a point of intersection with the x-axis, known as the x-intercept, and a point of intersection with the y-axis, known as the y-intercept. These key features allow for the analysis and interpretation of the function's behavior.
• Describe the relationship between the degrees of the numerator and denominator polynomials in a linear-over-linear rational function and its end behavior.
  • The end behavior of a linear-over-linear rational function is determined by the relative degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, the function will approach 0 as the input approaches positive or negative infinity. If the degree of the numerator is greater than the degree of the denominator, the function will approach positive or negative infinity as the input approaches positive or negative infinity. If the degrees of the numerator and denominator are equal, the function will approach a non-zero constant as the input approaches positive or negative infinity.
• Explain how transformations of linear-over-linear rational functions can be used to analyze and sketch their graphs.
  • Transformations of linear-over-linear rational functions, such as shifts, reflections, and scalings, can be used to analyze and sketch their graphs. By understanding how these transformations affect the key features of the graph, such as the asymptotes, intercepts, and overall shape, students can more effectively interpret and sketch the graphs of linear-over-linear rational functions. Transformations can also be used to model real-world situations involving inverse variation or rates of change, which are often represented by linear-over-linear rational functions.

Related terms

Rational Function: A rational function is a function that can be expressed as the quotient of two polynomial functions.

Polynomial Function: A polynomial function is a function where the variable is raised to a non-negative integer power.

First-Degree Polynomial: A first-degree polynomial is a polynomial where the highest exponent of the variable is 1.