Asymptotes And Graphing Of F(x) = 5x / (x^2 - 9)

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This article will guide you through the process of identifying the asymptotes of the rational function f(x) = 5x / (x^2 - 9) and then using that information to sketch its graph. Understanding asymptotes is crucial for analyzing the behavior of rational functions, especially as x approaches infinity or certain finite values. Asymptotes are lines that the graph of a function approaches but never quite touches, and they provide valuable insights into the function's overall shape and behavior. This function, f(x) = 5x / (x^2 - 9), presents an excellent opportunity to explore vertical, horizontal, and even oblique asymptotes, enhancing our understanding of rational functions and their graphical representations. By systematically identifying these asymptotes, we can accurately plot the function and interpret its key characteristics, including its domain, range, and behavior near points of discontinuity.

1. Identifying Vertical Asymptotes

To find the vertical asymptotes, we need to determine the values of x for which the denominator of the rational function is equal to zero, as these are the points where the function is undefined. In our case, the denominator is x² - 9. Setting this equal to zero gives us:

x² - 9 = 0

This is a simple quadratic equation that can be factored as follows:

(x - 3)(x + 3) = 0

Thus, the solutions are x = 3 and x = -3. These values indicate that there are vertical asymptotes at x = 3 and x = -3. Vertical asymptotes occur where the function approaches infinity (or negative infinity) as x approaches these values. Understanding vertical asymptotes is critical because they define the points where the function's graph will have sharp, almost vertical changes. These asymptotes serve as boundaries that the graph will approach but never cross, providing essential information about the function's behavior near these critical points. They also help in determining the function's domain, as the function is undefined at these x-values. The presence of vertical asymptotes significantly impacts the overall shape and interpretation of the rational function's graph, making their identification a key step in the analysis.

2. Determining Horizontal Asymptotes

To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive and negative infinity. This involves comparing the degrees of the numerator and the denominator. Our function is f(x) = 5x / (x² - 9). The degree of the numerator (5x) is 1, and the degree of the denominator (x² - 9) is 2.

When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0. This is because, as x becomes very large (either positive or negative), the denominator grows much faster than the numerator, causing the entire fraction to approach zero. The horizontal asymptote provides insight into the function's long-term behavior, showing how the function stabilizes as x moves towards infinity. Understanding the horizontal asymptote is crucial for sketching the graph accurately, as it indicates the level the function approaches but may not necessarily cross. In the context of this function, f(x) = 5x / (x² - 9), the horizontal asymptote at y = 0 signifies that the graph will get closer and closer to the x-axis as x becomes very large or very small.

3. Checking for Oblique (Slant) Asymptotes

An oblique asymptote (also known as a slant asymptote) exists when the degree of the numerator is exactly one greater than the degree of the denominator. In our function, f(x) = 5x / (x² - 9), the degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the denominator is not exactly one less than the numerator, there is no oblique asymptote for this function. Oblique asymptotes provide additional information about the function's behavior as x approaches infinity, showing a linear trend that is neither horizontal nor vertical. If an oblique asymptote were present, it would indicate that the function's graph approaches a slanted line as x goes to positive or negative infinity. The absence of an oblique asymptote in this case simplifies our analysis, as we only need to focus on the vertical and horizontal asymptotes to understand the function's asymptotic behavior. Recognizing when an oblique asymptote is present is important for accurately sketching the graph and understanding the function's long-term trends.

4. Determining the Asymptotes

Based on our analysis:

  • Vertical Asymptotes: x = 3 and x = -3
  • Horizontal Asymptote: y = 0
  • Oblique Asymptote: None

Therefore, the correct asymptotes are x = 3, x = -3, and y = 0. These asymptotes are critical for understanding the behavior of the function and for accurately sketching its graph. The vertical asymptotes define the points where the function is undefined and where the graph will have sharp vertical changes. The horizontal asymptote indicates the function's long-term behavior as x approaches infinity. Without oblique asymptotes, the function's asymptotic behavior is fully described by its vertical and horizontal asymptotes. By identifying these asymptotes, we can gain a clear understanding of the function's key characteristics and how it behaves across its domain. This information is essential for creating an accurate graphical representation and for interpreting the function's properties.

5. Graphing the Function

Now that we've identified the asymptotes, we can use this information to graph the function f(x) = 5x / (x² - 9). The asymptotes serve as guidelines for sketching the curve. Here's how we can approach the graphing process:

  1. Plot the Asymptotes: First, draw the vertical asymptotes at x = 3 and x = -3 as dashed vertical lines. Then, draw the horizontal asymptote at y = 0 (the x-axis) as a dashed horizontal line. These lines will help define the regions where the graph can exist and how it behaves as it approaches these boundaries. Plotting asymptotes first is crucial for creating an accurate sketch, as they provide the framework for the function's behavior. This step ensures that the graph correctly reflects the function's undefined points and its long-term trends.

  2. Find Key Points: Choose some test values for x in the intervals defined by the vertical asymptotes (i.e., x < -3, -3 < x < 3, and x > 3) and calculate the corresponding f(x) values. These points will help you understand the shape of the curve in each region. Key points include where the function intersects the x-axis (y = 0) and y-axis (x = 0), as well as any local maxima or minima. Calculating these points provides crucial anchors for the graph, ensuring it accurately represents the function's behavior. For instance, plotting the x-intercepts helps in understanding where the function changes sign, while the y-intercept gives the function's value at x = 0. Using these key points, you can start to sketch the curve within the bounds set by the asymptotes.

  3. Determine Intercepts: Find the x-intercept(s) by setting f(x) = 0 and solving for x. Find the y-intercept by setting x = 0 and evaluating f(0). Intercepts provide critical points where the graph crosses the axes, offering valuable reference points for sketching the curve. The x-intercepts, in particular, indicate where the function's sign changes, which is important for understanding the function's behavior across its domain. Similarly, the y-intercept shows the function's value at the origin. Calculating and plotting intercepts helps in creating a more accurate and detailed graph, as they highlight significant characteristics of the function.

    • X-intercepts: 5x / (x² - 9) = 0 implies 5x = 0, so x = 0 is the x-intercept.
    • Y-intercept: f(0) = 5(0) / (0² - 9) = 0, so y = 0 is the y-intercept.

  4. Sketch the Graph: Using the asymptotes and the key points, sketch the graph. Remember that the graph will approach the asymptotes but never cross them (unless it's a horizontal asymptote, which the graph can cross). Sketching the graph involves connecting the key points in a way that respects the asymptotes and reflects the overall behavior of the function. In the intervals between vertical asymptotes, the graph will either increase or decrease, approaching the asymptotes as x goes to infinity or the asymptote's value. The horizontal asymptote indicates how the graph behaves as x becomes very large or very small. By carefully considering these factors, you can create an accurate representation of the function's graph, showing its key features and behavior across its domain.

  • For x < -3, the function is negative and approaches the asymptotes x = -3 and y = 0.
  • For -3 < x < 0, the function is positive and approaches the asymptotes x = -3 and x = 3.
  • For 0 < x < 3, the function is negative and approaches the asymptotes x = 3 and x = -3.
  • For x > 3, the function is positive and approaches the asymptotes x = 3 and y = 0.

6. Conclusion

By systematically identifying the vertical and horizontal asymptotes and plotting key points, we can accurately graph the rational function f(x) = 5x / (x² - 9). The vertical asymptotes at x = 3 and x = -3, along with the horizontal asymptote at y = 0, provide a framework for understanding the function's behavior. Understanding asymptotes is essential for analyzing rational functions and for sketching their graphs. Asymptotes define the boundaries of the function's behavior, indicating where the function approaches infinity or a specific value. The combination of asymptotes and key points allows for a comprehensive understanding of the function's shape and trends. Mastering the techniques for finding and interpreting asymptotes is a valuable skill in calculus and mathematical analysis. By following the steps outlined in this article, you can confidently analyze and graph rational functions, gaining insights into their unique characteristics and behavior.

The correct answer for the asymptotes is:

D. x = 3, x = -3, y = 0