Graph Crossing The X-axis Root Of F(x)=(x+4)^6(x+7)^5

JU07/08/2025 08, 2025 by THE IDEN 54 views

At Which Root Does the Graph of f(x)=(x+4)^6(x+7)^5 Cross the x-axis?

Introduction

In the realm of mathematics, particularly in the study of functions and their graphical representations, identifying the points where a graph intersects the x-axis is a fundamental concept. These points, known as the roots or zeros of the function, hold significant importance in understanding the behavior and characteristics of the function. In this article, we will delve into the process of determining the roots of a given function and specifically address the question: At which root does the graph of f(x) = (x+4)^6(x+7)5 cross the x-axis?

Understanding the concept of roots and their relationship to the x-axis is crucial for solving this problem. Roots are the values of x for which the function f(x) equals zero. Graphically, these are the points where the curve intersects the x-axis. The key to finding the roots of a polynomial function lies in factoring the expression and setting each factor equal to zero. This is because if any factor is zero, the entire product becomes zero, satisfying the condition f(x) = 0. Our function f(x) = (x+4)^6(x+7)5 is already presented in a factored form, which simplifies our task significantly. We have two factors: (x+4)^6 and (x+7)^5. Each of these factors can potentially contribute to the roots of the function.

To determine the roots, we will set each factor equal to zero and solve for x. For the factor (x+4)^6, setting it to zero gives us (x+4)^6 = 0. Taking the sixth root of both sides, we get x+4 = 0, which leads to x = -4. Similarly, for the factor (x+7)^5, setting it to zero gives us (x+7)^5 = 0. Taking the fifth root of both sides, we get x+7 = 0, which leads to x = -7. Therefore, the roots of the function f(x) = (x+4)^6(x+7)5 are x = -4 and x = -7. However, the question asks at which root the graph crosses the x-axis, not just touches it. This introduces the concept of multiplicity, which we will explore further to answer the question accurately.

Multiplicity and Graph Behavior

To accurately determine where the graph of f(x) = (x+4)^6(x+7)5 crosses the x-axis, we need to understand the concept of multiplicity. Multiplicity refers to the number of times a particular root appears as a solution to the polynomial equation. In simpler terms, it is the exponent of the factor that gives rise to the root. The multiplicity of a root has a direct impact on how the graph of the function behaves at that point. Specifically, it determines whether the graph crosses the x-axis or simply touches it and turns around.

In our function, f(x) = (x+4)^6(x+7)5, the root x = -4 comes from the factor (x+4)^6. The exponent of this factor is 6, which means the root x = -4 has a multiplicity of 6. Similarly, the root x = -7 comes from the factor (x+7)^5. The exponent of this factor is 5, so the root x = -7 has a multiplicity of 5. The multiplicity of a root dictates the behavior of the graph at that x-value. A key principle to remember is that if a root has an even multiplicity, the graph will touch the x-axis at that point and turn around, without crossing it. Conversely, if a root has an odd multiplicity, the graph will cross the x-axis at that point.

Considering the multiplicities of our roots, we can now analyze the graph's behavior at x = -4 and x = -7. The root x = -4 has a multiplicity of 6, which is an even number. Therefore, at x = -4, the graph of f(x) will touch the x-axis and turn around, without crossing it. On the other hand, the root x = -7 has a multiplicity of 5, which is an odd number. This means that at x = -7, the graph of f(x) will cross the x-axis. This distinction is crucial for answering the question of where the graph crosses the x-axis. By understanding the concept of multiplicity and its effect on graph behavior, we can accurately identify the root where the graph transitions from one side of the x-axis to the other.

Determining the Crossing Root

Having established the significance of multiplicity in determining graph behavior at roots, we can now pinpoint the root at which the graph of f(x) = (x+4)^6(x+7)5 crosses the x-axis. As discussed earlier, the roots of this function are x = -4 and x = -7. The multiplicity of the root x = -4 is 6, which is an even number. This indicates that the graph touches the x-axis at x = -4 but does not cross it. Instead, the graph will approach the x-axis, touch it, and then turn back in the direction it came from. This behavior is characteristic of roots with even multiplicities.

Conversely, the root x = -7 has a multiplicity of 5, which is an odd number. This signifies that the graph crosses the x-axis at x = -7. At this point, the graph transitions from being below the x-axis to above it (or vice versa). The crossing behavior is a hallmark of roots with odd multiplicities. Therefore, based on the multiplicities of the roots, we can conclude that the graph of f(x) = (x+4)^6(x+7)5 crosses the x-axis at x = -7. This understanding of multiplicity allows us to not only find the roots of a function but also to predict the behavior of its graph around those roots.

To further illustrate this concept, consider the graph of a simple quadratic function with a repeated root, such as g(x) = (x-2)^2. This function has a root at x = 2 with a multiplicity of 2. The graph of this function is a parabola that touches the x-axis at x = 2 and bounces back, never crossing the x-axis. In contrast, a function like h(x) = (x-3)^3 has a root at x = 3 with a multiplicity of 3. The graph of this function crosses the x-axis at x = 3, demonstrating the characteristic crossing behavior associated with odd multiplicities. These examples reinforce the principle that the multiplicity of a root is a key determinant of how the graph interacts with the x-axis.

Conclusion

In conclusion, the graph of the function f(x) = (x+4)^6(x+7)5 crosses the x-axis at the root x = -7. This determination is based on the concept of multiplicity, which dictates the behavior of a graph at its roots. Roots with even multiplicities cause the graph to touch the x-axis and turn around, while roots with odd multiplicities cause the graph to cross the x-axis. The root x = -7 has a multiplicity of 5, which is odd, thus the graph crosses the x-axis at this point. Understanding the relationship between roots, multiplicities, and graph behavior is fundamental in analyzing and interpreting polynomial functions. This knowledge allows us to accurately predict how a graph will interact with the x-axis and provides valuable insights into the function's overall characteristics. By carefully considering the multiplicities of roots, we can gain a deeper understanding of the graphical representation of functions and their mathematical properties.