Problem 75 - Algebra and Trigonometry

Question

Graph the rational function \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2} $$

Step-by-Step Solution

Answer

The vertical asymptote is at \(x = 2\). Both \(f(x)\) and \(g(x)\) exhibit the same end behavior: \(f(x)\) behaves like \(x^2\).

Step 1: Identify the vertical asymptotes of f(x)

The vertical asymptotes occur at the values of \(x\) that make the denominator of \(f(x)\) equal to zero. The denominator is \(x - 2\), so set \(x - 2 = 0\) to find the vertical asymptote. Solving this equation gives \(x = 2\). Thus, there is a vertical asymptote at \(x = 2\).

Step 2: Simplify the rational function f(x)

To graph \(f(x)\), we first need to see if the rational function simplifies. Check if \(x - 2\) is a factor of the numerator \(x^3 - 2x^2 + 16\). Perform polynomial long division or use synthetic division to simplify it. If \(x - 2\) is not a factor, use the division result for correct polynomial representation for better graphing.

Step 3: Determine the end behavior of f(x) and g(x)

The degree of the numerator of \(f(x)\) is 3, and the degree of the denominator is 1, meaning the rational function will behave like \(x^2\) for large values of \(x\). The function \(g(x) = x^2\) also has a degree of 2. So, both have the same end behavior, dominated by the largest power term \(x^2\) as \(x\) approaches infinity or negative infinity.

Step 4: Graph f(x) and g(x)

Plot the function \(f(x) = \frac{x^3 - 2x^2 + 16}{x - 2}\), showing a vertical asymptote at \(x = 2\). Then, plot \(g(x) = x^2\). Use a graphing calculator or software to ensure they both display a similar trend for large values of \(x\), confirming their end behavior is the same. Ensure the viewing rectangle is large enough to clearly demonstrate how both graphs align as \(x\) approaches infinity or negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a rational function are like invisible lines where the graph shoots off indefinitely. They can be found by determining the values of \( x \) that make the denominator zero. In the function \( f(x) = \frac{x^3 - 2x^2 + 16}{x - 2} \), the denominator is \( x - 2 \).
By setting \( x - 2 = 0 \), we find \( x = 2 \). This means there is a vertical asymptote at \( x = 2 \).

• Vertical asymptotes are never crossed by the graph.
• They mark the boundary where the function behaves very erratically.

You identify these by examining the denominator, making it an important step in graphing rational functions.

End Behavior

The end behavior of a function describes what happens to the values of a function as \( x \) approaches infinity or negative infinity. For rational functions, this often depends on the degree of the numerator and the denominator.

In our example, for \( f(x) = \frac{x^3 - 2x^2 + 16}{x - 2} \), the numerator has a degree of 3, and the denominator a degree of 1. This implies the end behavior is driven by the term \( x^2 \).

Compare this with the function \( g(x) = x^2 \), where the highest degree is also 2. Hence, both functions approach similar values as \( x \) becomes very large or very small.

• For a rational function where the numerator's degree is higher than the denominator's, expect a polynomial-like end behavior.
• End behavior helps predict what the graph will look like far from the origin.

Polynomial Division

Polynomial division, such as long division or synthetic division, is essential for simplifying rational functions. By simplifying \( f(x) \), it becomes easier to plot and understand.

In the given \( f(x) = \frac{x^3 - 2x^2 + 16}{x - 2} \), you would perform polynomial division to see how the terms reduce.
If \( x-2 \) is not a factor, then the output provides you a precise representation for graphing.

• Long division is similar to numeric division but works with polynomial terms.
• Simplification aids in identifying other features of the graph, such as holes or lower-degree terms.

Understanding and applying polynomial division can greatly enhance the accuracy of your graphing efforts.

Graphing Functions

Graphing a rational function involves showcasing its behavior visually to interpret its characteristics better. Begin by identifying roots, vertical asymptotes, and end behavior from the algebraic expression.

For \( f(x) = \frac{x^3 - 2x^2 + 16}{x - 2} \), plot points around the vertical asymptote \( x = 2 \) and observe the end behavior as \( x \) moves towards infinity. For \( g(x) = x^2 \), simply plot the classic parabola.

Use graphing calculators or software to assist in illustrating these trends accurately.
Ensure your graphing window is wide enough to display similarities that confirm the end behavior of both functions.

• Highlight important features such as symmetry, intercepts, and asymptotes.
• Graphs offer a clear and precise method to visualize mathematical behavior.

By graphing, you apply theoretical concepts such as asymptotes and end behavior, turning them into tangible, visual forms.