Graphing F(x) = X³ - 3x - 2 And Finding Double Roots

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In this article, we will delve into the process of graphing the cubic function f(x) = x³ - 3x - 2. We will explore the key features of the graph, including its intercepts, turning points, and end behavior. A crucial aspect of our analysis will be the identification of double roots, which are points where the graph touches the x-axis but does not cross it. Understanding double roots is essential for comprehending the complete behavior of polynomial functions. Through a combination of algebraic techniques and graphical analysis, we aim to provide a comprehensive understanding of the function and its properties. This exploration will not only enhance our understanding of cubic functions but also provide valuable insights into the broader realm of polynomial functions and their applications. Mastering these concepts is crucial for success in advanced mathematics and related fields, where polynomial functions play a fundamental role in modeling and problem-solving. Let's embark on this journey to unravel the intricacies of f(x) = x³ - 3x - 2 and its graphical representation.

To effectively graph the function f(x) = x³ - 3x - 2, we need to follow a systematic approach that involves identifying key features and plotting strategic points. This process allows us to accurately represent the function's behavior and understand its characteristics. Let's break down the steps involved:

1. Finding the Intercepts:

  • Y-intercept: The y-intercept is the point where the graph intersects the y-axis. To find it, we set x = 0 in the function: f(0) = (0)³ - 3(0) - 2 = -2. Thus, the y-intercept is (0, -2). This point provides a crucial anchor for our graph, indicating where the function begins its journey on the coordinate plane.
  • X-intercepts (Roots): The x-intercepts, also known as roots or zeros, are the points where the graph intersects the x-axis. To find them, we set f(x) = 0 and solve for x: x³ - 3x - 2 = 0. This cubic equation can be solved by factoring. By observation or using the Rational Root Theorem, we can find that x = -1 is a root. Synthetic division or polynomial long division can then be used to divide the cubic by (x + 1). The result is a quadratic equation that can be factored further or solved using the quadratic formula. Factoring, we get: (x + 1)(x² - x - 2) = (x + 1)(x + 1)(x - 2) = 0. This gives us roots at x = -1 (with multiplicity 2) and x = 2. The multiplicity of the root at x = -1 indicates that the graph will touch the x-axis at this point but not cross it, a key characteristic of a double root.

2. Determining Turning Points:

Turning points, also known as local maxima and minima, are points where the graph changes direction. These points are crucial for understanding the function's shape and overall behavior. To find turning points, we use calculus:

  • First Derivative: Find the first derivative of the function, f'(x). For f(x) = x³ - 3x - 2, the first derivative is f'(x) = 3x² - 3. The first derivative gives us information about the slope of the tangent line at any point on the graph. Setting f'(x) = 0 allows us to find critical points, which are potential locations for turning points.
  • Critical Points: Set f'(x) = 0 and solve for x: 3x² - 3 = 0 implies x² = 1, so x = ±1. These are our critical points. At these x-values, the tangent line to the graph is horizontal, indicating a potential change in direction.
  • Second Derivative: Find the second derivative of the function, f''(x). For f'(x) = 3x² - 3, the second derivative is f''(x) = 6x. The second derivative provides information about the concavity of the graph. A positive second derivative indicates that the graph is concave up, while a negative second derivative indicates that the graph is concave down.
  • Concavity and Turning Points: Evaluate f''(x) at the critical points: f''(-1) = -6 (concave down, local maximum) and f''(1) = 6 (concave up, local minimum). This confirms that we have a local maximum at x = -1 and a local minimum at x = 1. The values of the function at these points are f(-1) = (-1)³ - 3(-1) - 2 = 0 and f(1) = (1)³ - 3(1) - 2 = -4. Thus, the turning points are (-1, 0) and (1, -4).

3. Analyzing End Behavior:

End behavior describes what happens to the function's values as x approaches positive and negative infinity. For polynomial functions, the end behavior is primarily determined by the leading term, which in this case is x³. As x approaches positive infinity, f(x) also approaches positive infinity. As x approaches negative infinity, f(x) also approaches negative infinity. This information helps us sketch the overall direction of the graph as it extends beyond the turning points and intercepts.

4. Plotting the Graph:

Using the information gathered, we can now plot the graph:

  • Plot the intercepts: (0, -2), (-1, 0), and (2, 0).
  • Plot the turning points: (-1, 0) (local maximum) and (1, -4) (local minimum).
  • Sketch the graph, considering the end behavior: The graph starts from negative infinity, rises to the local maximum at (-1, 0), touches the x-axis but does not cross it (due to the double root), falls to the local minimum at (1, -4), and then rises towards positive infinity, crossing the x-axis at (2, 0).

By following these steps, we can create an accurate and informative graph of the function f(x) = x³ - 3x - 2. The graph visually represents the function's behavior, including its roots, turning points, and end behavior. This graphical representation is invaluable for understanding the function's properties and solving related problems.

Based on the graph and our previous calculations, we can clearly identify the double root of the function f(x) = x³ - 3x - 2. A double root occurs at a point where the graph touches the x-axis but does not cross it. This is a characteristic feature of roots with even multiplicity. In our case, the factored form of the equation, (x + 1)²(x - 2) = 0, reveals that x = -1 is a root with multiplicity 2, making it a double root.

The graph of the function visually confirms this. At x = -1, the graph touches the x-axis and changes direction without crossing it, indicating a double root. This behavior is in contrast to the root at x = 2, where the graph crosses the x-axis. Understanding the concept of double roots is crucial in polynomial analysis as it provides insights into the function's behavior and the nature of its roots. Double roots often signify points of tangency between the graph and the x-axis, representing a unique characteristic of polynomial functions with repeated factors. In this context, x = -1 is indeed the double root of the given function.

As we have established through both algebraic analysis and graphical representation, the double root of the function f(x) = x³ - 3x - 2 occurs at x = -1. This conclusion is supported by the factored form of the equation, (x + 1)²(x - 2) = 0, which clearly shows that (x + 1) is a repeated factor. The graphical representation further reinforces this finding, as the graph touches the x-axis at x = -1 but does not cross it, a distinctive trait of double roots.

Therefore, among the given options:

  • A. -2
  • B. -1
  • C. 1
  • D. 2

the correct value for x that represents the double root of the function is B. -1. This value is not just a root of the function; it is a root with a multiplicity of 2, signifying its unique role in shaping the function's graph and behavior. Identifying double roots is a critical skill in polynomial analysis, enabling us to fully understand the nature and characteristics of polynomial functions.

In conclusion, we have successfully graphed the function f(x) = x³ - 3x - 2 and identified its double root. Through a step-by-step process, we found the intercepts, determined the turning points using calculus, and analyzed the end behavior of the function. This comprehensive approach allowed us to accurately sketch the graph and visually confirm the existence of a double root. The algebraic analysis, specifically factoring the equation, further supported our findings, revealing that x = -1 is a root with multiplicity 2.

The double root at x = -1 is a crucial feature of the function, influencing its behavior around this point. The graph touches the x-axis at x = -1 but does not cross it, a characteristic trait of roots with even multiplicity. Understanding the concept of double roots is essential for a thorough understanding of polynomial functions and their graphical representations. This analysis demonstrates the interplay between algebraic techniques and graphical interpretation in mathematics, highlighting the importance of both approaches in solving problems and gaining insights into mathematical concepts. By mastering these skills, we can tackle more complex problems and deepen our understanding of the mathematical world.