Graph each rational function. f(x)=(x^4-20 x^2+64)/(x^4-10 x^2+9)

Name: Problem 99, Chapter 3: Student's Solutions Manual for College Algebra
Uploaded: 2021-12-03T20:58:19-08:00
Duration: 5 min 48 s
Channel: Eric Carlsen
Description: Graph each rational function. f(x)=(x^4-20 x^2+64)/(x^4-10 x^2+9)

step 1 All right, we've been asked to graph this function, and what an absolutely ugly function it is.

Graph each rational function. f(x)=(x^4-20 x^2+64)/(x^4-10...

Key Concepts

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. They are central to algebra and calculus due to their unique behavior, combining polynomial features with discontinuities and asymptotic behavior. Analyzing these functions involves studying their domain, intercepts, and asymptotic limits.

Domain and Discontinuities

The domain of a rational function includes all real numbers except those values that make the denominator zero. Points where the denominator equals zero must be examined further to determine whether they result in vertical asymptotes or removable discontinuities (holes) if the factor cancels with a corresponding term in the numerator.

Factorization and Simplification

Factoring the numerator and the denominator is a crucial step in analyzing rational functions. It can help reveal common factors that simplify the expression, indicate potential removable discontinuities, and assist in identifying the zeros and vertical asymptotes of the function.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of the rational function becomes zero (and the corresponding factors don't cancel out). These asymptotes represent lines that the graph approaches but never touches, as the function's value tends to infinity or negative infinity near these points.

Horizontal/Oblique Asymptotes

The behavior of rational functions for large absolute values of x is determined by the degrees of the polynomials in the numerator and denominator. When the degrees are equal, the horizontal asymptote is the quotient of the leading coefficients. If the numerator has a lower degree than the denominator, the horizontal asymptote is y = 0. If the numerator’s degree is higher (often by one), an oblique asymptote may be present, representing a slanted line that the graph approximates at extreme values of x.

Zeros of a Rational Function

The zeros of a rational function are the solutions to the equation formed by setting the numerator equal to zero, provided these zeros do not cancel with factors in the denominator. They correspond to the points where the graph crosses or touches the x-axis, reflecting the roots of the numerator polynomial.

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