Problem 72 - Precalculus Mathematics for Calculus

Question

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{2 x^{3}+2 x}{x^{2}-1}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 1\) and \(x = -1\); slant asymptote at \(y = 2x\). Graph accordingly.

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function is equal to zero, provided the numerator is not also zero at those points. Set the denominator equal to zero and solve for \(x\): \(x^2 - 1 = 0\). Factoring, we have \((x+1)(x-1)=0\). Thus, the vertical asymptotes occur at \(x=1\) and \(x=-1\).

Step 2: Determine the Slant Asymptote

A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. Here, the numerator, \(2x^3 + 2x\), is degree 3 and the denominator, \(x^2-1\), is degree 2. Perform polynomial long division of \(2x^3 + 0x^2 + 2x\) by \(x^2 - 1\) to find the slant asymptote. Dividing, the quotient is \(2x\), so the slant asymptote is \(y = 2x\).

Step 3: Sketch the Graph

With the vertical asymptotes at \(x=1\) and \(x=-1\), and the slant asymptote at \(y=2x\), you can sketch the graph. For more accuracy, consider additional properties such as intercepts: the y-intercept is found by evaluating the function at \(x=0\), which gives \(r(0) = 0\). Then, choose several points around the asymptotes to see how the function behaves as \(x\) approaches these lines.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes are lines where the function tends to infinity. They occur when the denominator of a rational function equals zero, but not the numerator. This is because division by zero is undefined, causing the function's value to spike upwards or downwards.
To find vertical asymptotes in the function \( r(x) = \frac{2x^3 + 2x}{x^2 - 1} \), we set the denominator \( x^2 - 1 \) equal to zero.
Solving \((x+1)(x-1)=0\) gives \(x=1\) and \(x=-1\). These are where the function will have vertical asymptotes.
Always remember that if the numerator was also zero at these points, it would not be a vertical asymptote.

Using Polynomial Long Division for Slant Asymptotes

A slant (or oblique) asymptote appears when the degree of the numerator is one higher than that of the denominator. It is a straight line that the curve approaches as \(x\) goes to infinity.
In our example, \(r(x) = \frac{2x^3 + 2x}{x^2 - 1}\), the numerator is of degree 3 and the denominator is degree 2, so a slant asymptote exists.
To find it, use polynomial long division of \(2x^3 + 2x\) by \(x^2 - 1\).
When dividing, the quotient \(2x\) gives us the slant asymptote \(y = 2x\).
Remember to ignore any remainders; they only affect the curve close to finite values of \(x\).

Graph Sketching Tips

Sketching a graph involves understanding how a function behaves near its asymptotes and intercepts. It's like a roadmap for the function's path.
For \(r(x)\) as provided, we know:

• Vertical asymptotes at \(x = 1\) and \(x = -1\).
• A slant asymptote at \(y = 2x\).

To complete a sketch:

• Plot the y-intercept, found by setting \(x = 0\), resulting in \(r(0) = 0\).
• Choose points on either side of the vertical asymptotes \(x = 1\) and \(x = -1\) to see how the function behaves.
• Watch the graph approach \(y = 2x\) as \(x\) heads towards positive or negative infinity.

Connecting these dots helps plot a clear and accurate graph.

Analyzing Function Behavior

Function behavior analysis provides insights into how the function acts at important points and asymptotes. The goal is to predict the curve's path and shape.
For \(r(x)\), consider:

• The trend around asymptotes as the curve goes to \(+\infty\) or \(-\infty\).
• How the presence of \(y = 2x\) influences the slope at large values of \(x\).

By analyzing intercepts, you get more puzzle pieces to complete the graph. If the intercepts, asymptotes, and additional points are plotted correctly, the function's shape becomes evident.
Remember, the change in curve near asymptotes can vary from vertical rises to gentle slopes, depending on adjacent function points.