Problem 76 - 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. $$ f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2} $$

Step-by-Step Solution

Answer

The vertical asymptote of \(f(x)\) is at \(x=1\), and both \(f(x)\) and \(g(x)\) have the same end behavior: both resemble \(-x^2\) as \(x\to\infty\).

Step 1: Determine the Vertical Asymptotes

To find vertical asymptotes of the function \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), set the denominator equal to zero, i.e., solve \((x-1)^2 = 0\). This gives \(x = 1\). Therefore, there is a vertical asymptote at \(x = 1\).

Step 2: Analyze the Function's Degree for End Behavior

The degree of the numerator \(-x^4 + 2x^3 - 2x\) is 4 and the degree of the denominator \((x-1)^2\) is 2. Therefore, the end behavior resembles \((-1) \times x^{4-2} = -x^2\). Hence, the rational function \(f(x)\) has the same end behavior as the polynomial function \(g(x) = 1 - x^2\).

Step 3: Graph the Function and Identify Features

Plot the function \(f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2}\) considering the vertical asymptote at \(x=1\) and the end behavior similar to \(g(x) = 1-x^2\). Throughout the graph, as \(x\) approaches positive or negative infinity, both functions \(f(x)\) and \(g(x)\) tend to resemble \(-x^2\). These shared features include the shape of a downward-opening parabola with alterations near the vertical asymptote for \(f(x)\).

Step 4: Compare and Confirm End Behavior

By graphing both \(f(x)\) and \(g(x)\) in a large viewing window, observe that as \(x\) tends to infinity or negative infinity, \(f(x)\) closely follows the behavior of \(-x^2\), mirrored by \(g(x)\) except for any variance close to the vertical asymptote. The end behavior of both functions is thus confirmed as identical far from disruptions caused by the asymptote.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never touches or crosses. They occur in rational functions where the denominator equals zero, resulting in a division by zero.
To find these asymptotes, focus on setting the denominator of the rational function to zero. For the function \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), you identify vertical asymptotes by solving the equation \((x-1)^2 = 0\).
This gives \(x = 1\), translating to a vertical asymptote at \(x = 1\).

• These asymptotes create a boundary that the function approaches but never crosses.
• As \(x\) approaches these values from either direction, the graph moves towards positive or negative infinity.
• Vertical asymptotes are represented as dashed lines in the graph.

Vertical asymptotes are essential for understanding the behavior of graphs near points of discontinuity.

End Behavior

End behavior describes how the function behaves as \(x\) approaches infinity or negative infinity. This is determined largely by the degrees of the polynomial expressions in the numerator and denominator of a rational function.
In the function \( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), notice the degrees:

• The numerator is a degree 4 polynomial \(-x^4 + 2x^3 - 2x\).
• The denominator is a degree 2 polynomial \((x-1)^2\).

For end behavior analysis, compare these:
1. **Numerator's Degree versus Denominator's Degree**: Here, the numerator's degree is greater, specifically 4 compared to 2, implying that as \(x\) becomes very large, the function behaves like \(-x^2\).
2. **Functional Comparison**: - \(f(x)\) mimics the polynomial \(-x^2\), akin to \(g(x) = 1-x^2\).
This means:

• Both functions trend downwards, resembling the shape of an inverted parabola.
• As \(x\) moves far away from zero, the influence of lower degree terms diminishes, reinforcing this parabolic shape.

End behavior is key in predicting the long-term trends of rational functions.

Polynomial Division

Polynomial division plays a significant role in simplifying expressions and understanding the behavior of rational functions. When assessing the end behavior or simplifying a function, dividing the polynomials in the numerator and denominator can clarify its structure.
In the function\( f(x) = \frac{-x^4 + 2x^3 - 2x}{(x-1)^2} \), the higher degree of the numerator over the denominator suggests using division to predict dominance.
This division allows us to see that:

• The function simplifies in terms of its highest degree, focusing on influential terms.
• As discussed, \(-x^2\) emerges, dictating the end behavior by overpowering lesser terms when \(x\) is large.

Polynomial division also reveals more about how functions interact, providing insights into intercepts and potential crossing points beyond just limits and asymptotes.
For students, mastering polynomial division is a vital tool for analyzing and interpreting complex rational expressions.