Finding Roots: Does The Graph Cross The X-Axis?

Hey everyone! Let's dive into a cool math problem involving the function f(x) = (x + 4)⁶(x + 7)⁵. We're going to figure out which roots of this function cause the graph to actually cross the x-axis. This is a super important concept in understanding how functions behave, so pay close attention, guys! It's all about where the graph touches or intersects that x-axis, and we'll break it down step by step to make it crystal clear. So, grab your pencils and let's get started. We will explore the characteristics of roots, specifically focusing on how they affect the behavior of a function's graph near the x-axis. The analysis involves understanding the concept of multiplicity and its impact on whether a function crosses or touches the x-axis. We will look at the mathematical properties of polynomial functions and how they determine the behavior of a graph around its roots. This will include an explanation of how to identify the roots of a polynomial function from its factored form and understanding the relationship between the exponent of a factor and the graph's behavior at the corresponding root. We'll also provide a detailed explanation, including examples, to solidify your comprehension and application of this concept. Keep in mind that understanding these principles is crucial for anyone studying algebra or calculus. It provides a foundational understanding necessary for more advanced mathematical concepts and applications. Are you ready? Let's get to it!

Decoding the Function: Roots and Exponents

Alright, let's break down the function f(x) = (x + 4)⁶(x + 7)⁵. The roots of a function are the x-values where the function equals zero. In simpler terms, it's where the graph of the function touches or crosses the x-axis. When a factor in the function becomes zero, the entire function becomes zero. So, to find the roots, we need to look at each factor and see when it equals zero. Specifically, we're interested in the x-values that make each part of the equation equal to zero. This is where the graph interacts with the x-axis. To start, let's examine the first part: (x + 4)⁶. To make this part zero, what value of x do we need? Well, if x + 4 = 0, then x = -4. This is one of our roots! The exponent of 6 tells us something important, but we'll get to that in a bit. Next, let's look at the second part of our function: (x + 7)⁵. Following the same logic, we need x + 7 = 0, which means x = -7. This is our other root. Understanding the exponents attached to each factor is key to determining how the graph behaves at each root. The exponent indicates the multiplicity of each root, which directly influences whether the graph crosses the x-axis or just touches it. Knowing these root behaviors will help us sketch the graph accurately and understand the overall shape of the function. Knowing what the root is is not enough. Let's dig deeper into the concept!

The roots, or zeros, are the values of x for which f(x) = 0. For our function, f(x) = (x + 4)⁶(x + 7)⁵, we have two factors that could potentially equal zero: (x + 4) and (x + 7). Setting each factor to zero, we find the roots: For (x + 4) = 0, the root is x = -4. For (x + 7) = 0, the root is x = -7. These are the x-values where the graph of the function will intersect the x-axis, or at least touch it. This is where it gets interesting, as it is only a start. The exponent of each factor tells us more about the behavior of the graph at each root. When a root has an odd multiplicity (like the exponent 5 in (x + 7)⁵), the graph crosses the x-axis at that point. When a root has an even multiplicity (like the exponent 6 in (x + 4)⁶), the graph touches the x-axis at that point but does not cross it. This distinction is crucial to accurately sketching the graph of a polynomial function. The multiplicity of a root is the number of times that root appears as a solution to the equation f(x) = 0. For example, the root x = -4 has a multiplicity of 6 because the factor (x + 4) is raised to the power of 6. The root x = -7 has a multiplicity of 5 because the factor (x + 7) is raised to the power of 5. The multiplicity of a root directly affects how the graph behaves at that point. We will investigate this concept in the next section!

Crossing vs. Touching: The Power of Multiplicity

Okay, so we've found our roots: x = -4 and x = -7. Now, let's talk about the super important concept of multiplicity. Multiplicity refers to how many times a particular root appears in the function. It's determined by the exponent of the factor. When a root has an odd multiplicity, the graph crosses the x-axis at that point. When a root has an even multiplicity, the graph touches the x-axis at that point but doesn't cross it – it just bounces off. This is super important to remember, you guys! In our function, the root x = -4 comes from the factor (x + 4)⁶. Because the exponent is 6 (an even number), the graph touches the x-axis at x = -4 but doesn't cross it. It kind of bounces off the x-axis there. On the other hand, the root x = -7 comes from the factor (x + 7)⁵. The exponent is 5 (an odd number), so the graph crosses the x-axis at x = -7. This is because an odd power, like the 5th power, ensures that the function changes sign as it passes through the root. It goes from negative to positive or vice versa. The even power, like 6, means the function doesn't change sign. So, it stays on the same side of the x-axis. Understanding multiplicity is key. Let's delve deeper into this concept.

Here’s how to think about it: imagine a ball rolling along the x-axis. If the ball encounters a root with odd multiplicity, it crosses the x-axis and continues on its way. If it encounters a root with even multiplicity, it bounces off the x-axis and changes direction. It's like the x-axis is a wall. This behavior is a direct consequence of the properties of polynomial functions. Odd multiplicities mean that as x moves past a root, the sign of the function changes. Even multiplicities mean the sign stays the same. The sign change dictates whether the graph crosses or just touches. This behavior is predictable and consistent for all polynomials. This knowledge is important, so let's summarize: If the exponent on a factor is odd, the graph crosses the x-axis. If the exponent on a factor is even, the graph touches but does not cross the x-axis. The power of multiplicity isn't just a mathematical quirk; it helps us to accurately sketch graphs of polynomial functions without having to plot a ton of points. By knowing the roots and their multiplicities, you can quickly determine the basic shape of the graph. It's like having a cheat sheet for graphing polynomials. We have to know the rules, which include the multiplicity of each root to accurately sketch the graph of our function. The concept of multiplicity extends to all polynomial functions, not just the one we're looking at today. It's a fundamental aspect of understanding how polynomials behave, so it's a valuable concept to learn. It provides a deeper insight into the relationship between the algebraic form of a function and its graphical representation. Alright! Let's get a summary of what we know, and then we will be done.

Conclusion: The Final Answer

Alright, let's recap, friends! In the function f(x) = (x + 4)⁶(x + 7)⁵, we've identified two roots: x = -4 and x = -7. The root x = -4 has an even multiplicity (6), so the graph touches the x-axis at this point. The root x = -7 has an odd multiplicity (5), so the graph crosses the x-axis at this point. So, to answer the question, the graph of the function f(x) = (x + 4)⁶(x + 7)⁵ crosses the x-axis at x = -7. Knowing how to find the roots and understand their behavior helps us understand polynomial functions! You did a great job!

This knowledge is essential for solving calculus problems and understanding more complex functions. Keep practicing, and you'll become pros at this! I hope this explanation has been helpful, guys. Understanding the behavior of graphs near the x-axis is crucial for a complete understanding of functions. Keep practicing and applying these concepts, and you will become skilled in the analysis of functions. Remember that these principles are not just for this problem but can be applied to countless other scenarios. Keep exploring and happy learning! With this, we've successfully navigated the graph of our function and understood its behavior. You've got this!