4.8
A rational function is the ratio of two polynomials, with a non-zero denominator.
A key feature of rational functions is their asymptotes—lines that the graph approaches but never touches.
These asymptotes can be vertical or horizontal.
Vertical asymptotes occur where the denominator equals zero, creating breaks in the graph. These are found by solving the denominator for zero.
Horizontal asymptotes describe the end behavior of a rational function as the input becomes very large or very small.
Rational functions may also cross horizontal asymptotes, unlike vertical ones, which act as strict boundaries.
The position of the horizontal asymptotes is determined by comparing the degrees of the numerator and denominator.
When the numerator’s degree is less than the denominator’s, the horizontal asymptote is y equals zero.
A greater degree in the numerator than the denominator means the function has no horizontal asymptote.
For instance, consider the concentration of a pollutant in water, modeled as a rational function. As water volume increases, the concentration of a pollutant decreases and approaches zero, forming a horizontal asymptote at y equals zero.
A rational function is defined as the quotient of two polynomials:
where Q(x)≠0, These functions often exhibit asymptotes, which are the lines that the graph approaches but never touches. These asymptotes are classified based on how the function behaves near specific values of the input.
Vertical asymptotes occur where the denominator is zero, and the numerator is not, causing the function to be undefined. These are found by solving Q(x)=0. For example:
has a vertical asymptote at x=3, where the graph diverges to infinity.
Horizontal asymptotes describe the function’s behavior as the input becomes very large in magnitude. Their position depends on the degrees of the numerator and denominator:
When the degree of the numerator is exactly one more than the degree of the denominator, a slant asymptote exists. It is found using polynomial division. For instance:
has the slant asymptote y = x - 1.
Rational functions can cross horizontal or slant asymptotes but never vertical ones.
A rational function is the ratio of two polynomials, with a non-zero denominator.
A key feature of rational functions is their asymptotes—lines that the graph approaches but never touches.
These asymptotes can be vertical or horizontal.
Vertical asymptotes occur where the denominator equals zero, creating breaks in the graph. These are found by solving the denominator for zero.
Horizontal asymptotes describe the end behavior of a rational function as the input becomes very large or very small.
Rational functions may also cross horizontal asymptotes, unlike vertical ones, which act as strict boundaries.
The position of the horizontal asymptotes is determined by comparing the degrees of the numerator and denominator.
When the numerator’s degree is less than the denominator’s, the horizontal asymptote is y equals zero.
A greater degree in the numerator than the denominator means the function has no horizontal asymptote.
For instance, consider the concentration of a pollutant in water, modeled as a rational function. As water volume increases, the concentration of a pollutant decreases and approaches zero, forming a horizontal asymptote at y equals zero.
From Chapter 4:
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