Graph Touch X-Axis: Find The Root Of F(x)

Nov 13, 2025 by Admin 42 views

Hey guys! Let's dive into a fun math problem where we need to figure out at which root the graph of the function f(x) = (x-5)^3(x+2)2 just touches the x-axis. This is a classic question that tests our understanding of polynomial functions and their graphs. So, grab your thinking caps, and let's get started!

Understanding the Function

First, let's break down the function f(x) = (x-5)^3(x+2)2. We see that it's a polynomial function expressed in factored form. The factors are (x-5)^3 and (x+2)^2. Each factor gives us a root of the function. Remember, roots are the values of x for which f(x) = 0. So, to find the roots, we set each factor equal to zero:

(x-5)^3 = 0 gives us x = 5
(x+2)^2 = 0 gives us x = -2

Thus, the roots of the function are x = 5 and x = -2. But here's the catch: the powers to which these factors are raised (the exponents) tell us something important about how the graph behaves at these roots. These exponents are called multiplicities.

The multiplicity of a root determines whether the graph crosses the x-axis at that root or just touches it and turns around (bounces off). If the multiplicity is odd, the graph crosses the x-axis. If the multiplicity is even, the graph touches the x-axis. This is a crucial concept for understanding the behavior of polynomial graphs. Now, let's analyze the multiplicities of our roots in f(x) = (x-5)^3(x+2)2.

For the root x = 5, the factor is (x-5)^3, so the multiplicity is 3, which is odd. This means the graph crosses the x-axis at x = 5. On the other hand, for the root x = -2, the factor is (x+2)^2, so the multiplicity is 2, which is even. Therefore, the graph touches the x-axis at x = -2. The multiplicity of a root plays a significant role in determining the behavior of the graph near that root, influencing whether it crosses or simply touches the x-axis. So, keep in mind that when we have an odd multiplicity, we can expect the graph to cut through the x-axis, while an even multiplicity will result in the graph just kissing the x-axis and turning around. This is super helpful for sketching polynomial functions accurately. In the context of graphing polynomial functions, understanding multiplicities is key to predicting how the graph will interact with the x-axis at each root. This understanding allows for more accurate sketches and a deeper comprehension of the function's behavior overall. In summary, by analyzing the multiplicities of the roots, we can predict whether the graph of a polynomial function will cross or touch the x-axis at each root, giving us valuable insights into the graph's shape and characteristics.

Determining Where the Graph Touches the x-axis

Okay, now that we've identified our roots (x = 5 and x = -2) and understood the concept of multiplicities, let's pinpoint where the graph of f(x) touches the x-axis. As we discussed, the graph touches the x-axis at roots with even multiplicity. Looking back at our function f(x) = (x-5)^3(x+2)2, we found that the root x = -2 has a multiplicity of 2 (because of the (x+2)^2 factor), which is even. Therefore, the graph of f(x) touches the x-axis at x = -2. Conversely, the root x = 5 has a multiplicity of 3 (due to the (x-5)^3 factor), which is odd, so the graph crosses the x-axis at x = 5. The behavior of a polynomial graph near its roots is dictated by the multiplicities of those roots. An even multiplicity causes the graph to touch the x-axis and turn around, while an odd multiplicity leads to the graph crossing the x-axis. Therefore, it is crucial to carefully examine the function and identify the multiplicities of each root to understand how the graph interacts with the x-axis.

When the graph touches the x-axis, it's like a brief encounter where the graph meets the axis and then immediately veers away without crossing over. In contrast, when the graph crosses the x-axis, it continues its trajectory through the axis, moving from one side to the other. So, the distinction between touching and crossing is significant in understanding the graph's behavior. In our case, f(x) touches the x-axis at x = -2 because the multiplicity of that root is even, indicating a turning point. It crosses the x-axis at x = 5, where the multiplicity is odd, showing the graph's continuation through the axis. Understanding these nuances helps in accurately sketching polynomial graphs and predicting their behavior around the roots. By identifying the roots and their multiplicities, we can construct a clearer picture of how the graph interacts with the x-axis, providing valuable insights into the function's characteristics and overall shape. Hence, the multiplicity of each root serves as a guide to understanding and sketching polynomial graphs effectively.

The Answer

So, based on our analysis, the graph of f(x) = (x-5)^3(x+2)2 touches the x-axis at x = -2. This corresponds to option A in the multiple-choice question.

Key Takeaways

Roots and Factors: The roots of a function are the values of x that make the function equal to zero. These roots correspond to the factors of the polynomial.
Multiplicity: The multiplicity of a root is the power to which its corresponding factor is raised. It determines how the graph behaves at that root.
Even Multiplicity: The graph touches the x-axis at roots with even multiplicity.
Odd Multiplicity: The graph crosses the x-axis at roots with odd multiplicity.

Understanding these concepts is super helpful for analyzing polynomial functions and their graphs. You'll be able to quickly sketch the graph of a polynomial by identifying its roots and their multiplicities. This skill will come in handy in calculus and other advanced math courses.

Additional Tips

Sketching: When sketching a polynomial graph, start by plotting the roots on the x-axis. Then, use the multiplicities to determine whether the graph crosses or touches the x-axis at each root. Finally, consider the leading coefficient of the polynomial to determine the end behavior of the graph.
Practice: The best way to master these concepts is to practice! Try graphing different polynomial functions and analyzing their roots and multiplicities. You can find plenty of examples online or in textbooks.
Tools: Use graphing calculators or online graphing tools to visualize the graphs of polynomial functions. This can help you solidify your understanding of how the multiplicities affect the graph's behavior.

Conclusion

So there you have it! The graph of f(x) = (x-5)^3(x+2)2 touches the x-axis at x = -2. Remember the key concepts of roots, multiplicities, and how they affect the graph's behavior, and you'll be a pro at analyzing polynomial functions in no time! Keep practicing, and you'll ace those math problems. Happy graphing, everyone!