Problem 44
Question
Use a graphing utility to graph the function. What do you observe about its asymptotes? $$f(x)=-\frac{x}{\sqrt{9+x^{2}}}$$
Step-by-Step Solution
Verified Answer
The function \(f(x)=-\frac{x}{\sqrt{9+x^{2}}}\) has a horizontal asymptote at \(y=0\).
1Step 1: Identify the function
The given function is \(f(x)=-\frac{x}{\sqrt{9+x^{2}}}\).
2Step 2: Graph the function
To graph the function, plug it into a graphing utility. As \(x\) approaches positive and negative infinity, observe how the function behaves.
3Step 3: Identify the asymptotes
An asymptote is a line that the graph of a function approaches as \(x\) tends to positive infinity or negative infinity. Pay attention to where the function seems to level off but never actually reach. In this case, the function should approach asymptotes at \(y=0\), which is the x-axis.
Key Concepts
Asymptotes in Rational FunctionsGraphing Utilities in AlgebraRational Function Behavior
Asymptotes in Rational Functions
Asymptotes play a key role when exploring rational functions. They are like invisible boundaries that the graph of a function approaches but never touches or crosses. There are vertical, horizontal, and oblique (slant) asymptotes, and identifying them helps predict the behavior of a function far from the origin.
In the given exercise, the function is \(f(x)=-\frac{x}{\sqrt{9+x^{2}}}\). When graphing rational functions, we look for values that make the function undefined for vertical asymptotes and observe the end behavior for horizontal or slant asymptotes. For \(f(x)\), there are no vertical asymptotes because the denominator \(\sqrt{9+x^{2}}}\) is always positive.
However, there's a horizontal asymptote at \(y=0\), which is the x-axis. This is because as \(x\) approaches infinity in either direction, \(\frac{x}{\sqrt{9+x^{2}}}\) approaches \(0\), indicating that the function levels off at this line. Remembering to consider both positive and negative infinity when looking for horizontal asymptotes is a helpful exercise improvement advice.
In the given exercise, the function is \(f(x)=-\frac{x}{\sqrt{9+x^{2}}}\). When graphing rational functions, we look for values that make the function undefined for vertical asymptotes and observe the end behavior for horizontal or slant asymptotes. For \(f(x)\), there are no vertical asymptotes because the denominator \(\sqrt{9+x^{2}}}\) is always positive.
However, there's a horizontal asymptote at \(y=0\), which is the x-axis. This is because as \(x\) approaches infinity in either direction, \(\frac{x}{\sqrt{9+x^{2}}}\) approaches \(0\), indicating that the function levels off at this line. Remembering to consider both positive and negative infinity when looking for horizontal asymptotes is a helpful exercise improvement advice.
Graphing Utilities in Algebra
Graphing utilities are invaluable tools for visualizing algebraic functions. When numbers and abstract symbols become challenging, a visual representation can offer clarity in understanding complex behaviors. In the context of the exercise, a graphing utility allows us to input the function \(f(x)\) and visually confirm its properties and asymptotic behavior.
For students, harnessing these tools helps verify solutions and fosters a deeper conceptual understanding. Modern utilities can easily plot intricate graphs that would be time-consuming to draw by hand. They also provide dynamic capabilities, such as zooming and tracing, to examine specific points and relationships within the function, making them ideal for practicing algebra and precalculus concepts.
Incorporating graphing utilities early in the learning process can demystify abstract concepts and is recommended as an exercise improvement strategy. Students benefit from observing how changing function parameters directly affects the graph's shape and asymptotes.
For students, harnessing these tools helps verify solutions and fosters a deeper conceptual understanding. Modern utilities can easily plot intricate graphs that would be time-consuming to draw by hand. They also provide dynamic capabilities, such as zooming and tracing, to examine specific points and relationships within the function, making them ideal for practicing algebra and precalculus concepts.
Incorporating graphing utilities early in the learning process can demystify abstract concepts and is recommended as an exercise improvement strategy. Students benefit from observing how changing function parameters directly affects the graph's shape and asymptotes.
Rational Function Behavior
Understanding the behavior of rational functions is pivotal in building a deeper comprehension of algebra. A rational function's behavior is influenced by its numerator and denominator. In this example, \(f(x)\) is affected by the square root in the denominator, which plays a significant role in how the function behaves as \(x\) increases or decreases.
The function's behavior near and far from the origin allows us to draw the graph accurately. As \(x\) grows large, the function levels off toward the horizontal asymptote \(y=0\), indicating that the values of \(f(x)\) get closer to \(0\), but will never actually reach it. This behavior is essential for predicting what happens in the function's extreme ends and is a cornerstone for graphing rational functions correctly.
As part of exercise improvement, consider analyzing the function's behavior at various intervals and comparing it with the graph generated by a utility. This will not only confirm your predictions but also deepen your understanding and intuition about rational functions.
The function's behavior near and far from the origin allows us to draw the graph accurately. As \(x\) grows large, the function levels off toward the horizontal asymptote \(y=0\), indicating that the values of \(f(x)\) get closer to \(0\), but will never actually reach it. This behavior is essential for predicting what happens in the function's extreme ends and is a cornerstone for graphing rational functions correctly.
As part of exercise improvement, consider analyzing the function's behavior at various intervals and comparing it with the graph generated by a utility. This will not only confirm your predictions but also deepen your understanding and intuition about rational functions.
Other exercises in this chapter
Problem 43
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(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts o
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