Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = (x + 3)/(x + 2)

Verified step by step guidance

1. Identify the vertical asymptote by setting the denominator equal to zero: solve \(x + 2 = 0\) to find \(x = -2\). This is where the function is undefined.
2. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both are linear (degree 1), the horizontal asymptote is \(y = \frac{1}{1} = 1\).
3. Find the x-intercept by setting the numerator equal to zero: solve \(x + 3 = 0\) to find \(x = -3\). This is where the graph crosses the x-axis.
4. Find the y-intercept by evaluating the function at \(x = 0\): \(y = \frac{0 + 3}{0 + 2} = \frac{3}{2}\). This is where the graph crosses the y-axis.
5. Sketch the graph using the intercepts and asymptotes. Plot the x-intercept at \((-3, 0)\), the y-intercept at \((0, \frac{3}{2})\), and draw the vertical asymptote at \(x = -2\) and the horizontal asymptote at \(y = 1\). The graph approaches these asymptotes but never touches them.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Understanding the behavior of rational functions involves analyzing their domain, intercepts, and asymptotic behavior, which are crucial for graphing.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe the end behavior of the function. Identifying these helps in sketching the graph accurately.

Dominant Terms

Dominant terms in a rational function are the terms with the highest degree in the numerator and denominator. They determine the function's end behavior and help in finding horizontal or oblique asymptotes. For y = (x + 3)/(x + 2), the dominant terms are x/x, indicating a horizontal asymptote at y = 1.