Asymptotes And Graphing F(x) = 5x^3 / (x^2 - 4) A Comprehensive Guide

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When delving into the realm of rational functions, asymptotes are crucial to comprehend. An asymptote is a line that a curve approaches but does not intersect. Understanding asymptotes is vital for sketching the graph of a rational function accurately. For the given function, f(x) = 5x³ / (x² - 4), we will explore three types of asymptotes: vertical, horizontal, and oblique (or slant) asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero, provided the numerator does not equal zero at the same point. To find the vertical asymptotes for f(x) = 5x³ / (x² - 4), we set the denominator equal to zero:

x² - 4 = 0

This equation can be factored as:

(x - 2)(x + 2) = 0

Thus, the solutions are x = 2 and x = -2. These are the vertical asymptotes of the function. At x = 2 and x = -2, the function approaches infinity (or negative infinity), indicating a vertical asymptote. Vertical asymptotes are critical because they define the points where the function is undefined and provide essential guidance for graphing the function's behavior near these points. The function will drastically change its direction or sign as it approaches these vertical lines, making them fundamental features of the graph. Understanding the behavior near these asymptotes helps in accurately sketching the graph and predicting function values. For instance, knowing that f(x) has vertical asymptotes at x = 2 and x = -2 allows us to anticipate that the graph will exhibit dramatic changes around these x-values, either shooting towards positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To determine the horizontal asymptotes, we compare the degrees of the numerator and the denominator. In the function f(x) = 5x³ / (x² - 4), the degree of the numerator (3) is greater than the degree of the denominator (2). When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there may be an oblique asymptote, which we will discuss next. Understanding the relationship between the degrees of the polynomials in the numerator and denominator is essential for identifying the presence and nature of asymptotes. This comparison allows us to predict the long-term behavior of the function, which is invaluable in various mathematical and real-world applications. For example, in modeling physical systems, horizontal asymptotes can represent the steady-state value that a system approaches over time.

Oblique (Slant) Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the rational function has an oblique asymptote. To find the oblique asymptote for f(x) = 5x³ / (x² - 4), we perform polynomial long division. Dividing 5x³ by (x² - 4) gives us:

 5x
------------------
x² - 4 | 5x³ + 0x² + 0x + 0
 - (5x³ - 20x)
------------------
 20x

The quotient is 5x, and the remainder is 20x. Thus, we can express f(x) as:

f(x) = 5x + (20x / (x² - 4))

As x approaches infinity, the term (20x / (x² - 4)) approaches zero. Therefore, the oblique asymptote is y = 5x. Oblique asymptotes are particularly useful in understanding the end behavior of rational functions, indicating the direction the graph will follow as x moves towards infinity or negative infinity. In many applications, oblique asymptotes can model trends or approximations, such as in economics where they might represent the long-term growth rate of a company. This asymptotic behavior provides a critical perspective on the overall nature of the function beyond its local fluctuations.

To graph the function f(x) = 5x³ / (x² - 4), we combine our knowledge of the asymptotes with additional information about the function, such as intercepts and test points.

Step 1: Identify Asymptotes

We have already determined the asymptotes:

  • Vertical asymptotes: x = 2 and x = -2
  • Oblique asymptote: y = 5x

Step 2: Find Intercepts

To find the x-intercept, we set f(x) = 0:

5x³ / (x² - 4) = 0

This implies 5x³ = 0, so x = 0. Thus, the x-intercept is at (0, 0).

To find the y-intercept, we set x = 0:

f(0) = 5(0)³ / (0² - 4) = 0

Thus, the y-intercept is also at (0, 0).

Step 3: Determine Test Points

We choose test points in the intervals determined by the vertical asymptotes and intercepts:

  • Interval (-∞, -2): Test x = -3 f(-3) = 5(-3)³ / ((-3)² - 4) = 5(-27) / (9 - 4) = -135 / 5 = -27
  • Interval (-2, 0): Test x = -1 f(-1) = 5(-1)³ / ((-1)² - 4) = -5 / (1 - 4) = -5 / -3 = 5/3
  • Interval (0, 2): Test x = 1 f(1) = 5(1)³ / ((1)² - 4) = 5 / (1 - 4) = 5 / -3 = -5/3
  • Interval (2, ∞): Test x = 3 f(3) = 5(3)³ / ((3)² - 4) = 5(27) / (9 - 4) = 135 / 5 = 27

Step 4: Sketch the Graph

Using the asymptotes, intercepts, and test points, we can sketch the graph. The graph approaches the vertical asymptotes at x = -2 and x = 2. It also approaches the oblique asymptote y = 5x as x goes to positive or negative infinity. The graph passes through the origin (0, 0) and exhibits the behavior indicated by the test points. Sketching the graph involves plotting the key features like asymptotes and intercepts first, then using test points to fill in the curves. The vertical asymptotes dictate the vertical behavior near x = -2 and x = 2, while the oblique asymptote guides the long-term trend. The intercepts provide anchor points on the axes, and the test points confirm the function's sign and direction in various intervals. This method allows for an accurate representation of the function’s behavior, highlighting critical characteristics and tendencies.

Step 5: Analyze the Graph

By plotting these points and considering the asymptotic behavior, we observe that the graph has branches that approach the vertical asymptotes and slant asymptote as expected. In the interval (-∞, -2), the function is negative and decreases rapidly towards negative infinity as x approaches -2 from the left. Between (-2, 0), the function is positive, reaching a local maximum before crossing the x-axis at the origin. In the interval (0, 2), the function is negative, approaching negative infinity as x approaches 2 from the left. Finally, in the interval (2, ∞), the function is positive and increases rapidly towards positive infinity as x increases, closely following the slant asymptote y = 5x. This complete analysis, integrating asymptote identification, intercepts, test points, and graphing techniques, provides a thorough understanding of the rational function f(x) = 5x³ / (x² - 4).

Correct Asymptotes

Based on our analysis, the correct asymptotes are:

  • Vertical asymptotes: x = 2 and x = -2
  • Oblique asymptote: y = 5x

Therefore, the correct answer is C. x = 2, x = -2, y = 5x.

In conclusion, understanding and identifying asymptotes is crucial for accurately graphing rational functions. By finding vertical, horizontal, and oblique asymptotes, intercepts, and using test points, we can sketch the graph and analyze the behavior of the function effectively. In this example, the function f(x) = 5x³ / (x² - 4) has vertical asymptotes at x = 2 and x = -2, and an oblique asymptote at y = 5x. This comprehensive approach provides a solid foundation for working with rational functions and their graphical representations.