Negative Reciprocals

Definition

Negative reciprocals are the inverse of positive numbers, where the sign is changed from positive to negative. They are commonly used in the context of linear functions to describe the relationship between variables and their graphical representations.

5 Must Know Facts For Your Next Test

1. Negative reciprocals are the reciprocals of positive numbers with a negative sign, and they are used to represent the inverse relationship between variables in linear functions.
2. The graph of a negative reciprocal function is a hyperbola that opens either upward or downward, depending on the sign of the reciprocal.
3. Negative reciprocals are often used to describe the relationship between the independent and dependent variables in linear functions, such as the relationship between time and distance, or the relationship between the size of an object and its volume.
4. The slope of a negative reciprocal function is always negative, indicating an inverse relationship between the variables.
5. Negative reciprocals are important in understanding the behavior of rational functions, which are functions that can be expressed as the ratio of two polynomial functions.

Review Questions

• Explain how negative reciprocals are used to describe the relationship between variables in linear functions.
  • Negative reciprocals are used to represent the inverse relationship between variables in linear functions. This means that as one variable increases, the other variable decreases at a constant rate. The graph of a negative reciprocal function is a hyperbola that opens either upward or downward, depending on the sign of the reciprocal. The slope of a negative reciprocal function is always negative, indicating this inverse relationship between the variables.
• Describe the characteristics of the graph of a negative reciprocal function.
  • The graph of a negative reciprocal function is a hyperbola that opens either upward or downward, depending on the sign of the reciprocal. The graph passes through the origin and has two asymptotes, one vertical and one horizontal. The slope of the function is always negative, indicating an inverse relationship between the variables. As one variable increases, the other variable decreases at a constant rate, resulting in a hyperbolic curve that approaches the asymptotes but never touches them.
• Analyze the role of negative reciprocals in the context of rational functions.
  • Negative reciprocals are important in understanding the behavior of rational functions, which are functions that can be expressed as the ratio of two polynomial functions. Rational functions can exhibit a variety of behaviors, including asymptotic behavior, which is closely related to the presence of negative reciprocals in the function. The negative reciprocal terms in the denominator of a rational function can lead to vertical asymptotes, while the positive reciprocal terms in the numerator can lead to horizontal asymptotes. Understanding the role of negative reciprocals in rational functions is crucial for analyzing their properties and behavior.

Related terms

Reciprocal: The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse relationship between two quantities.

Inverse Function: An inverse function is a function that 'undoes' the original function, reversing the relationship between the input and output variables.

Linear Equation: A linear equation is an equation that represents a straight line on a coordinate plane, where the variables are raised to the first power and the equation is of the form $y = mx + b$.