Graph each rational function. See Examples 5–9. ƒ(x)=(18+6x-4x^2)/(4+6x+2x^2)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and overall shape, which are influenced by the degrees and coefficients of the polynomials.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator equals zero, indicating points of discontinuity. Horizontal asymptotes describe the behavior of the function as x approaches infinity, determined by the degrees of the numerator and denominator. Identifying these asymptotes is crucial for accurately graphing the function.

Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts occur when f(x) = 0, which means the numerator of the rational function must equal zero. The y-intercept is found by evaluating f(0). Identifying these intercepts helps in sketching the graph and understanding the function's behavior at specific points.