Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x-3}{x-2} $$

Short answer

The function has a y-intercept at -1.5 and x-intercept at 3, a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. It does not display any symmetry to neither the x-axis nor the y-axis. With these points and asymptotes we can sketch the graph of the function.

Step-by-step solution

Identify the Intercepts

The y-intercept is the value of y when x = 0. If you substitute x = 0 in the equation \(y = \frac{x-3}{x-2}\), the y-intercept y = -3/2. The x-intercept is the value of x for which y = 0. If you substitute y = 0 in the equation, you get x = 3 as the x-intercept.

Identify the Asymptotes

To find the vertical asymptote, set the denominator equal to zero and solve for x, which gives x = 2. This implies that the line x = 2 is a vertical asymptote. As for the horizontal asymptote, since the degree of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the coefficients of the highest powers of x in the numerator and denominator, or y = 1.

Analyze the Symmetry

Since neither replacing \(-x\) for x nor replacing \(-y\) for y results in the original equation, there is no symmetry to neither the x-axis nor the y-axis. Thus, the function is neither even nor odd.

Graph the function

Using the y-intercept (-1.5), x-intercept (3), vertical asymptote (x = 2), and horizontal asymptote (y = 1), sketch the graph. The graph should have two distinct parts divided by the vertical asymptote, approaching the asymptotes in the limit, but never touching them. Use a graphing utility to verify your result.