Numerator

Definition

The numerator is the top number in a fraction, which represents the quantity or number of parts being considered. It is an essential component of rational functions, as it helps determine the behavior and characteristics of the function.

5 Must Know Facts For Your Next Test

1. The numerator determines the overall behavior and shape of a rational function, such as the number and location of asymptotes.
2. The degree of the numerator polynomial compared to the degree of the denominator polynomial affects the end behavior of the rational function.
3. The sign of the numerator can determine the direction of the function's graph and the location of any vertical asymptotes.
4. Factoring the numerator can reveal important information about the function, such as the x-intercepts and any removable discontinuities.
5. Simplifying the numerator by canceling common factors with the denominator can help in sketching the graph of a rational function.

Review Questions

• Explain how the degree of the numerator polynomial compared to the degree of the denominator polynomial affects the end behavior of a rational function.
  • The degree of the numerator polynomial compared to the degree of the denominator polynomial determines the end behavior of a rational function. If the degree of the numerator is less than the degree of the denominator, the function will approach zero as the input approaches positive or negative infinity, resulting in a horizontal asymptote. 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, resulting in a slant asymptote. If the degrees are equal, the function will approach a non-zero constant value as the input approaches positive or negative infinity, resulting in a horizontal asymptote.
• Describe how the sign of the numerator can affect the direction of a rational function's graph and the location of any vertical asymptotes.
  • The sign of the numerator can have a significant impact on the direction of a rational function's graph and the location of any vertical asymptotes. If the numerator is positive, the function will be positive for all values of the input, and the graph will be above the x-axis. Conversely, if the numerator is negative, the function will be negative for all values of the input, and the graph will be below the x-axis. Additionally, the sign of the numerator can determine the location of any vertical asymptotes. If the numerator changes sign, the function will have a vertical asymptote at the value where the numerator is equal to zero.
• Analyze how factoring the numerator can reveal important information about the properties of a rational function.
  • Factoring the numerator of a rational function can provide valuable insights into the function's properties. By factoring the numerator, you can identify the x-intercepts of the function, as they correspond to the zeros of the numerator polynomial. This information can be used to sketch the graph of the function and determine its behavior. Additionally, factoring the numerator can reveal any removable discontinuities in the function, which occur when the numerator and denominator share a common factor. Recognizing and addressing these removable discontinuities can help in understanding the function's domain and behavior.

Related terms

Denominator:

The denominator is the bottom number in a fraction, which represents the total number of equal parts the whole is divided into.

Rational Expression:

A rational expression is a fraction where both the numerator and denominator are polynomials.

Rational Function:

A rational function is a function that can be written as the ratio of two polynomial functions.