Numerator

Definition

The numerator is the top part of a fraction, representing the number of parts considered from the whole. It plays a crucial role in defining the value of rational functions by indicating how many parts of the whole are being counted, relative to the denominator. In rational functions, the numerator helps to express polynomial relationships and is key to understanding their behavior as variables change.

5 Must Know Facts For Your Next Test

1. The numerator can be a constant, a variable, or a polynomial expression that represents various values in a rational function.
2. In analyzing rational functions, understanding the behavior of the numerator helps determine points where the function may be zero, which can indicate x-intercepts.
3. When simplifying rational functions, factoring the numerator is often necessary to identify common factors with the denominator.
4. The degree of the numerator in relation to the degree of the denominator affects the horizontal asymptotes of the rational function.
5. Changing the coefficients or terms in the numerator can significantly alter the graph of a rational function and its properties.

Review Questions

• How does the numerator affect the overall value and behavior of a rational function?
  • The numerator directly influences the value of a rational function as it determines what part of the whole is being represented. When analyzing graphs, it’s crucial because it shows where the function may reach zero (x-intercepts) and affects how the function behaves at infinity. If you change the numerator while keeping the denominator constant, it will change where the function intersects the x-axis and can lead to different characteristics in terms of increasing or decreasing intervals.
• Discuss how factoring the numerator impacts simplifying rational functions and finding common factors.
  • Factoring the numerator allows for simplification of rational functions by identifying any common factors with the denominator. This simplification can reveal additional insights about the function's behavior, such as removable discontinuities. When you factor both parts and cancel common factors, it often makes it easier to analyze limits and asymptotic behavior of the rational function.
• Evaluate how changes in the numerator affect vertical and horizontal asymptotes in rational functions.
  • Changes in the numerator can have profound effects on both vertical and horizontal asymptotes in rational functions. The vertical asymptotes are influenced by values where the denominator equals zero, while horizontal asymptotes are determined by comparing degrees between numerator and denominator. If you increase or alter terms in the numerator without changing its degree relative to the denominator, it can shift where horizontal asymptotes lie or even remove them entirely if there’s no longer a limit as x approaches infinity.

Related terms

Denominator:

The denominator is the bottom part of a fraction, which indicates into how many equal parts the whole is divided.

Rational Function: A rational function is a function defined by the ratio of two polynomials, where the numerator and denominator are both polynomials.

Polynomial: A polynomial is a mathematical expression that consists of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication.