Graph each rational function. See Examples 5–9.
$\text{ƒ}\left(x\right)=\left[\right(x+6\left)\right(x-2\left)\right]/\left[\right(x+3\left)\right(x-4\left)\right]$

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{(x+6)(x-2)}{(x+3)(x-4)}\). This function is a ratio of two polynomials.
2. Find the domain by determining the values of \(x\) that make the denominator zero. Set the denominator equal to zero: \((x+3)(x-4) = 0\), so \(x = -3\) and \(x = 4\) are excluded from the domain.
3. Find the zeros of the function by setting the numerator equal to zero: \((x+6)(x-2) = 0\). Solve for \(x\) to find the x-intercepts, which are \(x = -6\) and \(x = 2\).
4. Determine the vertical asymptotes by using the values excluded from the domain, which are \(x = -3\) and \(x = 4\). These are the vertical lines where the function is undefined.
5. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Both numerator and denominator are degree 2 polynomials, so the horizontal asymptote is the ratio of the leading coefficients. Since both leading coefficients are 1, the horizontal asymptote is \(y = 1\).

Key Concepts

Rational Functions and Their Domains

A rational function is a ratio of two polynomials. Its domain includes all real numbers except where the denominator equals zero, causing vertical asymptotes or holes. Identifying these restrictions is essential before graphing.

Asymptotes of Rational Functions

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, indicating values the function cannot take. Horizontal or oblique asymptotes describe end behavior as x approaches infinity, found by comparing degrees of numerator and denominator.

Intercepts and Critical Points

x-intercepts are found by setting the numerator equal to zero, while y-intercepts are found by evaluating the function at x=0. These points help plot the graph accurately and understand the function's behavior near the axes.