Graph each rational function. ƒ(x)=(18+6x-4x²)/(4+6x+2x²)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{18 + 6x - 4x^2}{4 + 6x + 2x^2}\). The goal is to analyze and graph this function step-by-step.
2. Rewrite the numerator and denominator in standard polynomial form by ordering terms from highest to lowest degree: Numerator: \(-4x^2 + 6x + 18\), Denominator: \$2x^2 + 6x + 4$.
3. Find the domain by determining where the denominator is zero. Solve the quadratic equation \$2x^2 + 6x + 4 = 0\( to find values of \)x$ that are not in the domain (vertical asymptotes or holes).
4. Determine horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator. Since both are degree 2, find the horizontal asymptote by dividing the leading coefficients: \(\frac{-4}{2}\).
5. Find the intercepts: For the y-intercept, evaluate \(f(0)\) by substituting \(x=0\) into the function. For x-intercepts, set the numerator equal to zero and solve \(-4x^2 + 6x + 18 = 0\) to find where the function crosses the x-axis.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values. Graphing involves analyzing behavior such as intercepts, asymptotes, and end behavior.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, indicating undefined points. Horizontal or oblique asymptotes describe the end behavior of the function as x approaches infinity or negative infinity.

Graphing Techniques for Rational Functions

Graphing involves finding intercepts by setting numerator or denominator to zero, determining asymptotes, and analyzing the function's behavior near these points. Plotting key points and understanding the function's limits help create an accurate graph.