Understanding Rational Functions | Graph Analysis, Asymptotes, Intercepts, and More

Rational Function (Graph)

A rational function is a function that can be written as the ratio of two polynomial functions

A rational function is a function that can be written as the ratio of two polynomial functions. In other words, it is of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials.

The graph of a rational function is the plot of the points (x, f(x)) that satisfy the equation f(x) = p(x) / q(x). The shape and behavior of the graph can provide information about the function’s domain, range, asymptotes, intercepts, and other characteristics.

To analyze the graph of a rational function, we consider the following:

1. Vertical asymptotes: Vertical asymptotes occur when the denominator of the rational function equals 0. These are vertical lines that the graph approaches but never crosses. To find the vertical asymptotes, we solve the equation q(x) = 0 and identify the x-values where the function is undefined.

2. Horizontal asymptotes: Horizontal asymptotes are the lines the graph approaches as x approaches positive or negative infinity. To determine the horizontal asymptotes, we analyze the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at the ratio of the leading coefficients. And if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

3. x-intercepts: The x-intercepts are the points where the graph intersects the x-axis, which occur when the numerator of the rational function equals 0. To find the x-intercepts, we solve the equation p(x) = 0 and identify the x-values where the function crosses the x-axis.

4. y-intercept: The y-intercept is the point where the graph intersects the y-axis, which occurs when x = 0. To find the y-intercept, we evaluate f(0) = p(0) / q(0).

5. Domain and range: The domain of a rational function consists of all possible x-values that make the function defined. We need to exclude any x-values that would make the denominator zero, as it would result in division by zero. The range is the set of all possible y-values that the function can output. It depends on the behavior of the graph and the presence of horizontal asymptotes.

By analyzing these aspects of the graph, we can gain a comprehensive understanding of the behavior and characteristics of a rational function.

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