Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{x^{3}}{x^{2}-1}$$

Step-by-Step Solution

Answer

The x-intercept is \(x = 0\). The vertical asymptotes are \(x = -1\) and \(x = 1\). The function has a slant asymptote at \(y = x\).

Step 1: Finding Intercepts

To find the x-intercepts, you set \(f(x) = 0\). Hence, the x-intercept is \(x = 0\). Set \(x = 0\) in the equation to find the y-intercept, which is also \(y = 0\).

Step 2: Determining Vertical Asymptotes

A vertical asymptote occurs where the function is undefined. Set the denominator of the function \(x^{2}-1 = 0\), we have \(x = -1,1\). So the vertical asymptotes are \(x = -1\) and \(x=1\).

Step 3: Determining Slant Asymptotes

Because the degree of the numerator exceeds the degree of the denominator by one, the function has a slant asymptote. Perform polynomial division or use the rule that the asymptote is the quotient of the leading terms. Thus, the slant asymptote is \(y = x\).

Step 4: Sketching the Graph

Use the above information to make a rough sketch of the function. The key features are: the function crosses the x-axis at the origin; the function is undefined at \(x = -1\) and \(x=1\), thus creating vertical asymptotes there; the curve approaches a slant line with slope 1 (that is, the line \(y = x\)) as \(x\) becomes large either positively or negatively.

Key Concepts

Finding Intercepts in Rational Functions

Understanding where a rational function intersects the axes is a crucial step in graphing the function. Intercepts are points where the graph crosses the x-axis (x-intercepts) or y-axis (y-intercepts). To find the x-intercepts, we look for values of x that make the function equal to zero, meaning the numerator of the rational function must be zero while the denominator is non-zero. In the exercise \(f(x)=\frac{x^{3}}{x^{2}-1}\), setting \(f(x) = 0\) leads to the conclusion that the x-intercept is at the origin, \(x = 0\).

For the y-intercept, we evaluate \(f(0)\), which implies plugging \(x = 0\) into the function. Given that both the numerator and the denominator of \(f(x)\) will be zero at this point, the y-intercept is also found to be at the origin, \(y = 0\). It is important to note that finding intercepts can vary in complexity depending on the rational function, and one should be careful to avoid undefined conditions, such as having a zero in the denominator. Intercepts provide a starting framework that aids in visualizing the overall shape of the graph.

Determining Vertical Asymptotes

Vertical asymptotes are lines to which a function's graph gets infinitely close as the independent variable either increases or decreases without bounds. They indicate where a function is undefined. For rational functions, vertical asymptotes commonly occur at values of x that result in a zero in the denominator. To find these critical values, we set the denominator of the function equal to zero and solve for x.

In the given function \(f(x)=\frac{x^{3}}{x^{2}-1}\), we determine the vertical asymptotes by finding the values that make \(x^{2}-1 = 0\). Solving this yields \(x = ±1\), so our function has vertical asymptotes at \(x = -1\) and \(x = 1\). These lines \(x = -1\) and \(x = 1\) represent the boundaries beyond which the graph of the function cannot cross. Understanding vertical asymptotes helps in predicting the behavior of the graph near these boundaries and is essential for sketching an accurate representation of the rational function.

Identifying Slant Asymptotes

Slant (or oblique) asymptotes occur when the rational function's numerator has a degree exactly one higher than its denominator. Unlike horizontal asymptotes, which remain at constant y-values, slant asymptotes have a slope and are represented by a linear equation. To find a slant asymptote, we typically perform polynomial long division or compare the leading terms of the numerator and denominator.

For the exercise at hand, \(f(x)=\frac{x^{3}}{x^{2}-1}\), the degree of the numerator (cubed term) exceeds the degree of the denominator (squared term) by exactly one. This indicates the presence of a slant asymptote. The slant asymptote can be found using the rule of leading terms, deriving \(y = x\) as the equation of the asymptote. As the value of x moves toward positive or negative infinity, the graph of \(f(x)\) will increasingly resemble the line \(y = x\), although it will never touch it. Identifying the slant asymptote provides a critical end behavior model for the rational function, assisting with accurate graphing, especially at the extremes of the x-axis.