A rational function is a type of function that can be expressed as the quotient of two polynomials, where the denominator is not zero. This type of function is important because it helps illustrate key concepts in understanding behaviors such as asymptotes, discontinuities, and overall graphing characteristics. Rational functions can model a variety of real-world scenarios, including rates, proportions, and more complex relationships between variables.
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Rational functions can have vertical asymptotes at values of x that make the denominator zero, indicating where the function is undefined.
The degree of the numerator and denominator polynomials determines the behavior of the rational function, including the presence and type of horizontal asymptotes.
A rational function can exhibit holes in its graph when a common factor exists in both the numerator and denominator.
As x approaches positive or negative infinity, the behavior of a rational function is influenced by the leading coefficients and degrees of the numerator and denominator.
Rational functions are used in various fields including physics, economics, and biology to represent relationships involving rates or ratios.
Review Questions
How do you determine the vertical asymptotes of a rational function?
To find the vertical asymptotes of a rational function, you need to identify the values of x that make the denominator equal to zero while ensuring these values do not also make the numerator zero. By setting the denominator polynomial equal to zero and solving for x, you can pinpoint where the function will be undefined and thus approach a vertical line in its graph.
Discuss how the degrees of the polynomials in a rational function affect its horizontal asymptotes.
The degrees of the numerator and denominator polynomials play a crucial role in determining horizontal asymptotes for a rational function. If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0. If they are equal, the horizontal asymptote can be found by taking the ratio of their leading coefficients. If the numerator's degree is greater than that of the denominator, there is no horizontal asymptote, though there may be an oblique asymptote.
Evaluate how discontinuities in a rational function affect its overall behavior and graphical representation.
Discontinuities in a rational function occur when there are values of x that make the denominator zero but do not correspond to zeroes in the numerator. These points create vertical asymptotes in the graph where the function tends to infinity or negative infinity. Additionally, if both polynomials share a common factor, this results in a hole on the graph. Understanding these discontinuities is vital for sketching accurate graphs and analyzing limits at specific points.
Related terms
Polynomial: A mathematical expression that consists of variables raised to whole number exponents and combined using addition, subtraction, and multiplication.
A line that a graph approaches as it heads toward infinity; rational functions often have vertical or horizontal asymptotes based on their polynomial degrees.