Problem 63 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{3}+1}{x^{2}+3 x}$$

Step-by-Step Solution

Answer

The rational function \(h(x)=\frac{x^{3}+1}{x^{2}+3x}\) has x-intercept of -1, y-intercept of 1/3, and vertical asymptotes at x = 0 and x = -3. There is no horizontal asymptote, but there is a slant asymptote at y = x - 3.

Step 1: Identify the Intercepts

The x-intercept is found by setting the numerator to zero and solving for x. Doing this gives \(x=-1\). The y-intercept is found by setting x=0 in the function. Doing this gives \(y=1/3\).

Step 2: Identify the Vertical Asymptote(s)

The vertical asymptote(s) are found by setting the denominator to zero and solving for x. Doing this yields \(x=0\) and \(x=-3\).

Step 3: Identify the Horizontal Asymptote

The horizontal asymptote is found by taking the limit of the function as x approaches positive and negative infinity. Because the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.

Step 4: Calculating the Slant Asymptote

We can divide the numerator by the denominator using polynomial long division. By doing that we find that the quotient (without the remainder) is \(x-3\), which is the equation of the slant (or oblique) asymptote.

Step 5: Sketch the Graph

Once all the intercepts and asymptotes are found, they can be plotted onto a graph. The function is now plotted with respect to the x and y intercepts, the vertical asymptotes, and the slant asymptote.

Key Concepts

Rational Function Intercepts

Understanding where a rational function intersects the axes is crucial for sketching its graph. Intercepts are the points where the function crosses the x-axis (x-intercepts) or y-axis (y-intercepts). To find the x-intercept, we set the numerator equal to zero and solve for x. In the given function, \(h(x)=\frac{x^{3}+1}{x^{2}+3 x}\), setting the numerator to zero, \(x^3 + 1 = 0\), gives us \(x=-1\) as the x-intercept.

For the y-intercept, we evaluate the function at \(x=0\). Substituting 0 into the function gives us a y-value of \(\frac{1}{3}\), making the y-intercept \(\left(0, \frac{1}{3}\right)\). Intercepts are essential for initial plotting before you draw the rest of the graph. It's like setting the starting point for your function on the graph before developing its shape around asymptotes.

Vertical Asymptotes

Vertical asymptotes are like boundaries that the graph of a rational function can never cross. They occur where the function is undefined—typically where the denominator equals zero. For our function, \(h(x)=\frac{x^{3}+1}{x^{2}+3 x}\), we find the vertical asymptotes by setting the denominator equal to zero and solving for x. This gives us two vertical asymptotes at \(x=0\) and \(x=-3\).

When graphing, we draw dashed lines at each of these x-values to denote where the function will shoot up to positive or negative infinity, but will never intersect these vertical lines. Knowing where vertical asymptotes lie helps in predicting the behavior of the function and understanding its limits.

Slant Asymptotes

In contrast to vertical and horizontal asymptotes, slant asymptotes occur when the rational function has a higher degree in the numerator than in the denominator. Instead of leveling off, the graph will follow a diagonal line at extreme values of x. To find a slant asymptote, we use polynomial long division to divide the numerator by the denominator.

For our exercise, dividing \(x^3 + 1\) by \(x^2 + 3x\) yields a slant asymptote with the equation \(y=x-3\). On a graph, this is represented as a dashed diagonal line that the function approaches but never reaches. As the function moves towards positive or negative infinity, it will become increasingly close to this slant asymptote, giving us a clear vision for the end behavior of our function.