Rational Function

Definition

A rational function is a type of function that can be expressed as the quotient of two polynomial functions, where the denominator is not equal to zero. These functions can have a variety of behaviors and features, including asymptotes, intercepts, and end behavior, which are essential in understanding their graphs and how they behave in different situations.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes at values that make the denominator equal to zero, creating points where the function is undefined.
2. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials; if the degrees are equal, the asymptote is the ratio of the leading coefficients.
3. The graph of a rational function can cross horizontal asymptotes if it has certain characteristics based on its degree and leading coefficients.
4. To find intercepts of a rational function, set the numerator equal to zero for the x-intercepts and evaluate the function at zero for the y-intercept.
5. Rational functions can exhibit holes in their graphs, which occur at points where both the numerator and denominator share a common factor that can be canceled out.

Review Questions

• How do you determine the vertical and horizontal asymptotes of a rational function?
  • To find vertical asymptotes, identify values that make the denominator zero while ensuring these values do not also make the numerator zero. For horizontal asymptotes, compare the degrees of the numerator and denominator: if they are equal, use the ratio of their leading coefficients; if the degree of the numerator is less than that of the denominator, the horizontal asymptote is at y=0. This analysis helps visualize key features of the graph.
• Explain how you would identify and describe any holes present in a rational function's graph.
  • Holes in a rational function's graph occur at x-values where both the numerator and denominator share a common factor. To find these holes, factor both polynomials and cancel out any common factors. The x-values corresponding to these factors represent locations where the function is undefined but does not create an asymptote. At these points, you would also want to find corresponding y-values to describe their impact on the overall graph.
• Analyze how changes in the coefficients of a rational function affect its graph and key features like intercepts and asymptotes.
  • Changing coefficients in a rational function can significantly impact its graph by altering its steepness, position, and intercepts. For instance, increasing coefficients in the numerator can raise or lower x-intercepts while modifying coefficients in the denominator may shift vertical asymptotes. Understanding these relationships allows for deeper insights into how the function behaves overall as well as predicting shifts in graphs based on adjustments made to its expression.

Related terms

Polynomial Function: A function that is represented by a polynomial expression, which consists of variables raised to whole number powers and combined using addition, subtraction, and multiplication.

Vertical Asymptote: A vertical line that represents a value where a rational function approaches infinity or negative infinity as the input approaches a certain value from either side.

Horizontal Asymptote: A horizontal line that indicates the behavior of a rational function as the input approaches positive or negative infinity, showing the function's long-term values.