Question

Find any (a) vertical, (b) horizontal, and (c) slant asymptotes of the graph of the function. Then sketch the graph of \(f\). $$f(x)=\frac{x^{2}-7 x+12}{x-3}$$

Short answer

The function \(f(x) = \frac{x^2 - 7x + 12}{x-3}\) has a vertical asymptote at \(x = 3\) and doesn't have any horizontal or slant asymptotes.

Step-by-step solution

Determine the vertical asymptote

The vertical asymptote of a function can be found by setting the denominator equal to zero and solving for x. So, \(x - 3 = 0\). After solving, \(x = 3\) is obtained. Therefore, the vertical asymptote of the function is \(x = 3\).

Check for horizontal or slant asymptotes

In a rational function, if the degree of the numerator is less than or equal to the degree of the denominator, then there is a horizontal asymptote. If the degree of the numerator is exactly one more than the degree of the denominator, then there is a slant asymptote. In the given function, the degree of the numerator and the denominator are both 1. Hence, there is no slant, but there could be a horizontal asymptote. Divide the quotient of the leading terms, that is \(\lim_{{x \to \infty}} f(x) = \frac{x}{1} = x\). Thus, there is no horizontal asymptote.

Sketch the graph of the function

Using the asymptotes, draw a rough sketch of the function. Include the vertical asymptote at \(x = 3\) and observe that there is no horizontal or slant asymptotes, meaning that the function will go to positive or negative infinity as \(x\) goes to positive or negative infinity respectively.

Key concepts

Vertical Asymptote

When you're working with rational functions, a vertical asymptote represents a line that the function approaches but never touches as it extends towards infinity in the y-direction. To find a vertical asymptote, look at the denominator of the function and set it equal to zero, solving for the variable x.

For example, consider the function \(f(x) = \frac{x^2 - 7x + 12}{x - 3}\). The denominator is \(x - 3\), so setting it to zero gives us \(x = 3\). That means \(x = 3\) is a line that the graph of our function will approach but never intersect, defining our vertical asymptote.

Horizontal Asymptote

Horizontal asymptotes indicate the value that the function approaches as \(x\) goes to positive or negative infinity. To find horizontal asymptotes, examine the degrees of the numerator and denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be the x-axis, or \(y = 0\).

If the degrees are equal, divide the leading coefficients of the numerator and the denominator. This will give you the horizontal asymptote. In our case with \(f(x)\), the degrees are equal, but interestingly, there is no horizontal asymptote, because when dividing the leading coefficients, the result is an x-term, not a constant, implying that the function's value will grow indefinitely.

Slant Asymptote

A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than that of the denominator in a rational function. It's a straight line that the graph approaches as \(x\) increases or decreases without bound.

You can find a slant asymptote by using long division or synthetic division to divide the numerator by the denominator, discarding any remainder. In the provided exercise, because the degrees of the numerator and the denominator polynomial are equal, no slant asymptote exists. Slant asymptotes are not present in functions where the degree of the numerator and denominator are equal or where the denominator's degree is higher.

Graphing Rational Functions

When graphing rational functions, start by identifying the asymptotes and any intercepts. Plot these on the graph first. The vertical asymptote will suggest points through which the function cannot pass, while the horizontal or slant asymptote will show the function's end behavior.

For our exercise's function, start by marking the vertical asymptote at \(x = 3\). Since there are no horizontal or slant asymptotes, the graph will rise or fall indefinitely in sections divided by the vertical asymptote. Use additional points derived from the function to plot its graph, ensuring you're representing the behavior near the vertical asymptote accurately and considering the function's end behavior at extreme x-values.