Problem 57 - Calculus A New Horizon

Question

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$\frac{x^{2}-2}{x}$$

Step-by-Step Solution

Answer

Vertical asymptote at \(x=0\), oblique asymptote at \(y=x\). The graph passes through \((\sqrt{2}, 0)\) and \((-\sqrt{2}, 0)\).

Step 1: Identify the Function

The given function is \( f(x) = \frac{x^2 - 2}{x} \). This is a rational function with a polynomial in the numerator and a monomial in the denominator.

Step 2: Find Vertical Asymptotes

Set the denominator equal to zero and solve for \(x\). For \(f(x) = \frac{x^2 - 2}{x}\), set \(x = 0\). Thus, there is a vertical asymptote at \(x = 0\).

Step 3: Find Horizontal or Oblique Asymptotes

Compare the degrees of the numerator and the denominator. Here, the degree of the numerator \(x^2\) is 2, and the degree of the denominator \(x\) is 1. Since the degree of the numerator is higher, there is no horizontal asymptote, but an oblique asymptote. Perform polynomial long division to find it.

Step 4: Perform Polynomial Long Division

Divide \(x^2 - 2\) by \(x\). The result is \(x - 2/x\). As \(x\) approaches infinity or negative infinity, the term \(-2/x\) approaches zero. Therefore, the oblique asymptote is \(y = x\).

Step 5: Sketch the Graph

Plot the vertical asymptote \(x = 0\) as a vertical dashed line, and the oblique asymptote \(y = x\) as a diagonal dashed line. Since \( f(x) = x - \frac{2}{x} \), sketch curves approaching the asymptotes. The graph will pass near points where \(x^2 - 2 = 0\), factoring gives \(x = \pm \sqrt{2}\). Include these points \((\sqrt{2}, 0)\) and \((-\sqrt{2}, 0)\) as part of the curve.

Key Concepts

Asymptotes

Asymptotes are critical features when analyzing rational functions. They are lines that the graph of a function approaches but never quite touches. In the world of rational functions, asymptotes can be vertical, horizontal, or oblique. Understanding these helps students predict the behavior of the graph far away from the origin in terms of both direction and steepness. Asymptotes serve as guidelines that provide valuable insights into the graph's layout, dictating how the graph behaves at the extremes of the coordinate plane.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, provided there is no common factor with the numerator. These are essentially vertical lines where the function heads towards positive or negative infinity. In our example, we set the denominator equal to zero: for the function \(f(x) = \frac{x^2 - 2}{x}\), setting \(x = 0\) gives the vertical asymptote \(x = 0\). This line indicates where the graph will climb indefinitely upwards or downwards as it approaches this x-value from both sides.

Oblique Asymptotes

Oblique asymptotes, sometimes called slant asymptotes, arise when the degree of the polynomial in the numerator is one higher than the degree in the denominator. Unlike vertical asymptotes, these are diagonal lines that the graph of a function approaches. In the function \(f(x) = \frac{x^2 - 2}{x}\), the numerator \(x^2\) (degree 2) is one degree higher than the denominator \(x\) (degree 1), so there is no horizontal asymptote, rather an oblique one. To find this, use polynomial long division. Dividing \(x^2 - 2\) by \(x\) gives \(x - \frac{2}{x}\), indicating the oblique asymptote is \(y = x\) as \(-\frac{2}{x}\) becomes negligible when \(x\) is large or small.

Graph Sketching

When sketching the graph of a rational function, asymptotes provide crucial guidelines. Start by sketching the vertical and oblique asymptotes as dashed lines. For \(f(x) = \frac{x^2 - 2}{x}\), mark \(x = 0\) as a vertical asymptote and \(y = x\) as an oblique asymptote. Next, plot important points like where \(f(x)\) crosses the x-axis or y-axis. In this case, solve \(x^2 - 2 = 0\) to find points \((\sqrt{2}, 0)\) and \((-\sqrt{2}, 0)\). Finally, draw curves approaching these asymptotes, ensuring the graph behaves accurately. It should get closer to the asymptotes as it extends towards infinity on either side, reflecting the function's behavior continuum.