Graph Touches X-Axis: Finding Roots Of F(x)=(x-5)^3(x+2)^2

Nov 26, 2025 by Alex Johnson 59 views

Finding Where the Graph of f(x)=(x-5)^3(x+2)^2 Touches the x-axis

avigating polynomial functions can sometimes feel like exploring a maze, especially when trying to visualize their graphs. One crucial aspect of understanding a polynomial's graph is identifying its roots, also known as x-intercepts. These are the points where the graph intersects or touches the x-axis. In this article, we'll dissect the function f(x) = (x-5)^3(x+2)2 to pinpoint exactly where its graph touches the x-axis. This involves understanding the concept of multiplicity and how it affects the behavior of the graph at each root. So, let's embark on this mathematical journey together and unravel the secrets hidden within this equation. Understanding the behavior of polynomial functions is essential not only in mathematics but also in various fields such as physics, engineering, and computer science, where they are used to model real-world phenomena. By mastering the techniques for analyzing roots and their multiplicities, you'll gain a deeper appreciation for the power and versatility of these mathematical tools. Moreover, this knowledge will enable you to tackle more complex problems and develop a stronger foundation in mathematical analysis.

Understanding Roots and Their Multiplicity

To accurately determine where the graph of f(x) = (x-5)^3(x+2)2 touches the x-axis, we first need to grasp the concept of roots and their multiplicity. Roots, in the context of polynomial functions, are the values of x that make the function equal to zero. They are the solutions to the equation f(x) = 0, and graphically, they represent the points where the function's graph intersects the x-axis. These points are crucial for understanding the overall behavior of the function, as they mark the transitions between positive and negative values of f(x).

Multiplicity, on the other hand, refers to the number of times a particular root appears as a factor in the polynomial. For example, in the function f(x) = (x-5)^3(x+2)2, the factor (x-5) appears three times, so the root x = 5 has a multiplicity of 3. Similarly, the factor (x+2) appears twice, giving the root x = -2 a multiplicity of 2. The multiplicity of a root significantly influences the behavior of the graph at that point. Specifically, it determines whether the graph crosses the x-axis, touches it and bounces back, or exhibits some other unique behavior. Understanding the multiplicity of roots is essential for accurately sketching the graph of a polynomial function and interpreting its properties.

Analyzing the Given Function: f(x)=(x-5)^3(x+2)2

Now, let's apply these concepts to our specific function, f(x) = (x-5)^3(x+2)2. By setting f(x) = 0, we can identify the roots of the function. The equation (x-5)^3 = 0 gives us the root x = 5, and the equation (x+2)^2 = 0 gives us the root x = -2. So, we have two distinct roots: 5 and -2. However, as we discussed earlier, the multiplicity of these roots plays a critical role in understanding the graph's behavior. The root x = 5 comes from the factor (x-5)^3, which means it has a multiplicity of 3. This odd multiplicity tells us that the graph will cross the x-axis at x = 5. In contrast, the root x = -2 comes from the factor (x+2)^2, indicating a multiplicity of 2. This even multiplicity means that the graph will touch the x-axis at x = -2 but will not cross it. Instead, the graph will “bounce” off the x-axis at this point. Therefore, by analyzing the roots and their multiplicities, we can gain valuable insights into how the graph of the function behaves around these points.

Determining Where the Graph Touches the x-axis

Based on our analysis, we've determined that the graph of f(x) = (x-5)^3(x+2)2 touches the x-axis at the root with an even multiplicity. In this case, that root is x = -2, which has a multiplicity of 2. This means that at x = -2, the graph will come down to the x-axis, touch it, and then turn back in the same direction, without crossing over to the other side. This behavior is characteristic of roots with even multiplicities and is an important feature to recognize when sketching polynomial graphs. On the other hand, the root x = 5, with its odd multiplicity of 3, indicates that the graph will cross the x-axis at this point. This distinction between roots with even and odd multiplicities is fundamental to understanding the overall shape and behavior of polynomial functions. Therefore, by carefully examining the multiplicities of the roots, we can accurately predict how the graph will interact with the x-axis at each intercept.

Conclusion: The Graph Touches the x-axis at x = -2

In conclusion, by analyzing the function f(x) = (x-5)^3(x+2)2, we've successfully identified that the graph touches the x-axis at the root x = -2. This determination was made by understanding the concept of multiplicity and how it affects the behavior of the graph at each root. The even multiplicity of 2 associated with the root x = -2 indicates that the graph will touch the x-axis and bounce back, rather than crossing it. This analysis provides a clear and concise method for understanding the behavior of polynomial functions and their graphs. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems and gain a deeper appreciation for the elegance and power of mathematics. Remember, the key to success in mathematics lies in understanding the fundamental concepts and applying them systematically to solve problems. And if you are interested in learning more about polynomial functions and their graphs, you can visit Khan Academy for additional resources and examples.