Problem 48 - Calculus of a Single Variable

Question

In Exercises 45–48, use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur? $$ h(x)=\frac{-x^{3}+x^{2}+4}{x^{2}} $$

Step-by-Step Solution

Answer

The slant asymptote of the function \(h(x)\) is \(-x+1\). As you zoom out of the graph, the function's curve will appear more linear because it is approaching its asymptote, which is a straight line, for increasingly large values of \(x\).

Step 1 - Perform Polynomial Division

As the degree of the numerator is greater than denominator by 1, gladly it is guaranteed that there will exist a slant asymptote. Remember, a slant asymptote can exist only when the degree of the numerator function is only one order greater than that of the denominator. Perform the polynomial division of the numerator with the denominator. You can use synthetic division or long division.

Step 2 - Division Result

Using long division method, the division \( \frac{-x^{3}+x^{2}}{x^{2}} \) yields \(-x+1\). Hence, \(-x+1\) is the slant asymptote of the function \(h(x)\). You can verify this by graphing both the function \(h(x)\) and its slant asymptote on the same set of axes. The graph of \(h(x)\) approaches the line given by \(y = -x + 1\) as \(x\) tends towards positive or negative infinity.

Step 3 - Graph Behavioral Analysis

The graph of the function \(h(x)\) will 'follow' the slant asymptote as \(x\) tends towards positive or negative infinity. As you zoom out, you will observe that the graph of \(h(x)\) and line \(y=-x+1\) will appear closer together. The curve of \(h(x)\) will seem to become more 'linear'. This happens because the curve is approaching the slant asymptote line for larger and smaller values of \(x\). As you increase the viewable span of \(x\), those larger (and smaller) values of \(x\) become more and more predominant.

Key Concepts

Polynomial Division

Polynomial division is a crucial technique used in finding the slant asymptote of a rational function like \( h(x) = \frac{-x^{3} + x^{2} + 4}{x^{2}} \). In this process, you divide the numerator by the denominator, revealing a simpler expression that describes the behavior of the function at infinity. Think of it like breaking down a complex fraction into something more manageable.

There are various methods to perform polynomial division, including long division and synthetic division. Long division is similar to the way you divide numbers, taking each term of the polynomial step by step. Synthetic division can often be quicker but has limitations based on the coefficients of the divisor.

The outcome of polynomial division helps us identify the slant asymptote when the degree of the numerator is exactly one higher than that of the denominator. This result models the path the graph will "straighten out" towards at far extremes, essentially guiding us on the graph's asymptotic behavior.

Degree of a Polynomial

Understanding the degree of a polynomial is key to determining the existence of slant asymptotes. The degree of a polynomial is the highest power of the variable in its expression. In \( h(x) = \frac{-x^{3} + x^{2} + 4}{x^{2}} \), the numerator \(-x^{3} + x^{2}\) has a degree of 3, while the denominator \(x^{2}\) has a degree of 2.

This difference in degrees indicates that the graph of the function will have a slant (or oblique) asymptote because the numerator's degree surpasses that of the denominator by exactly one. Slant asymptotes provide a line that the graph of the function will approach but never cross entirely at infinity. Keeping track of polynomial degrees can help predict certain characteristics of the graph without needing to calculate extensively.

Graphing Utility

Graphing utilities are powerful tools for visualizing functions. They provide a dynamic way to understand the behavior of functions like \( h(x) = \frac{-x^{3} + x^{2} + 4}{x^{2}} \). A graphing calculator or software allows students to input a function and see its visual representation.

By observing the graph, students can identify important features such as intercepts, asymptotes, and end behaviors instantly. For our function, a graphing utility can help spot the slant asymptote \( y = -x + 1 \), showing how both the curve and the asymptote draw closer with zooming out.

• Graphing utilities enable easy manipulation of the view, offering insights into how the function behaves at varying scales.
• They are particularly useful for confirming mathematically derived details, like seeing that \( h(x) \) behaves similarly to its asymptote when \( x \) is very large or small.

Utilizing these resources builds a more intuitive understanding of complex functions.

Behavior at Infinity

A fundamental question in mathematics is understanding how functions behave as variables grow indefinitely large or small, known as behavior at infinity. This concept is crucial when dealing with rational functions and slant asymptotes.

For the function \( h(x) = \frac{-x^{3} + x^{2} + 4}{x^{2}} \), the charted path of the curve can be understood in terms of how \( h(x) \) approaches the slant asymptote \( y = -x + 1 \) as \( x \) goes towards infinity or negative infinity. Essentially, because the significant terms in both the numerator and denominator become dominant at these extremes, the graph will closely align with the slant asymptote line.

• This "straightening" effect happens because the higher degree terms overpower smaller terms, simplifying the expression.
• Visualizing this on a graphing tool, students see the graph become linear-like, solidifying the concept that at infinity, only leading terms matter.

Thus, behavior at infinity is about distilling a function down to its core tendencies, integral for comprehending how graphs align with asymptotes over vast scales.