Graph each rational function. ƒ(x)=[(x-5)(x-2)]/(x²+9)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{(x-5)(x-2)}{x^2 + 9}\). Notice that the numerator is a quadratic expression and the denominator is \(x^2 + 9\).
2. Determine the domain of the function by finding values of \(x\) that make the denominator zero. Since \(x^2 + 9 = 0\) has no real solutions, the domain is all real numbers.
3. Find the zeros of the function by setting the numerator equal to zero: \((x-5)(x-2) = 0\). Solve for \(x\) to find the x-intercepts.
4. Analyze the end behavior by considering the degrees of the numerator and denominator. Both numerator and denominator are degree 2, so find the horizontal asymptote by dividing the leading coefficients.
5. Plot key points including the x-intercepts, y-intercept (found by evaluating \(f(0)\)), and sketch the graph using the horizontal asymptote and the shape determined by the sign of \(f(x)\) in different intervals.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the form helps identify domain restrictions and behavior, such as vertical asymptotes where the denominator is zero.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. Vertical asymptotes occur at these excluded values, indicating where the function grows without bound or decreases without bound.

Graphing Rational Functions

Graphing involves finding intercepts, asymptotes, and behavior near asymptotes. For f(x) = [(x-5)(x-2)]/(x^2+9), since the denominator x^2+9 is never zero, there are no vertical asymptotes, and the graph’s shape is influenced by zeros of the numerator and end behavior.