Horizontal Asymptote

A horizontal asymptote is a horizontal line that a function approaches as the input values become very large (positive or negative). It is a key concept in understanding the behavior of functions as they approach infinity or negative infinity.

Finding Horizontal Asymptotes

To find the horizontal asymptote of a function, follow these steps:

Determine the degree of the numerator and denominator of the function.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = (leading coefficient of the numerator) / (leading coefficient of the denominator).
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Examples

Let's look at a few examples to better understand horizontal asymptotes:

Example 1:

Find the horizontal asymptote of the function f(x) = (3x^2 + 2) / (2x - 1).

Solution:

The degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Example 2:

Find the horizontal asymptote of the function g(x) = (4x^3 - 2x + 1) / (2x^3 + 5).

Solution:

The degree of the numerator is 3 and the degree of the denominator is also 3. Therefore, the horizontal asymptote is y = (4/2) = 2.

Summary

Understanding horizontal asymptotes is important in analyzing the long-term behavior of functions. By following the steps to find horizontal asymptotes, you can determine how a function behaves as the input values approach infinity or negative infinity.

◂Math Worksheets and Study Guides Eighth Grade. Linear equations

The resources above cover the following skills:

Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Geometry: Given a line in a coordinate plane, students understand that all 'slope triangles' - triangles created by a vertical 'rise' line segment (showing the change in y), a horizontal 'run' line segment (showing the change in x), and a segment of the line itself - are similar. They also understand the relationship of these similar triangles to the constant slope of a line.