X-Axis Crossing Points: F(x)=(x+4)^6(x+7)^5

Let's dive into finding out where the graph of the function f(x) = (x+4)^6 (x+7)^5 crosses the x-axis. This involves understanding the concept of roots and their multiplicity, and how they influence the behavior of the graph around those points. So, buckle up, and let's get started!

Understanding Roots and Multiplicity

Before we pinpoint where the graph crosses the x-axis, it's crucial to understand what roots are and the significance of their multiplicity. Roots, also known as zeros or x-intercepts, are the values of x for which the function f(x) equals zero. In simpler terms, they are the points where the graph of the function intersects or touches the x-axis. For our function, f(x) = (x+4)^6 (x+7)^5, we can identify the roots by setting each factor to zero.

The roots are x = -4 and x = -7. But here’s where it gets interesting: each root has a multiplicity, which is the power to which the corresponding factor is raised. The multiplicity tells us how the graph behaves near that particular root. For x = -4, the factor (x+4) is raised to the power of 6, so the multiplicity of the root x = -4 is 6. Similarly, for x = -7, the factor (x+7) is raised to the power of 5, so the multiplicity of the root x = -7 is 5.

Now, what does the multiplicity tell us? A root with an even multiplicity (like x = -4 with multiplicity 6) means that the graph touches the x-axis at that point but does not cross it. Instead, the graph bounces off the x-axis. This is because the function's sign doesn't change as x passes through the root. Think of a parabola touching the x-axis at its vertex – that's a classic example of a root with even multiplicity. On the other hand, a root with an odd multiplicity (like x = -7 with multiplicity 5) means that the graph crosses the x-axis at that point. The function's sign changes as x passes through the root, causing the graph to go from below the x-axis to above it (or vice versa).

Determining Where the Graph Crosses the X-Axis

Based on our understanding of roots and multiplicity, we can now determine where the graph of f(x) = (x+4)^6 (x+7)^5 crosses the x-axis. We have two roots: x = -4 with multiplicity 6 and x = -7 with multiplicity 5. Since the multiplicity of the root x = -4 is even, the graph touches the x-axis at x = -4 but does not cross it. The graph bounces off the x-axis at this point.

However, the multiplicity of the root x = -7 is odd. This means that the graph crosses the x-axis at x = -7. So, the answer to the question "At which root does the graph of f(x) = (x+4)^6 (x+7)^5 cross the x-axis?" is x = -7. To further illustrate this, imagine plotting the graph. As you approach x = -7 from the left, the function values are negative (the graph is below the x-axis). As you move past x = -7, the function values become positive (the graph is above the x-axis). This change in sign is what signifies a crossing.

In summary, the concept of multiplicity is essential for understanding the behavior of a graph near its roots. Even multiplicity implies a touch (no crossing), while odd multiplicity implies a crossing. Therefore, by analyzing the roots and their multiplicities, we can accurately determine where a graph crosses the x-axis.

Graphical Representation

To solidify our understanding, let's consider a graphical representation of the function f(x) = (x+4)^6 (x+7)^5. While I can't physically draw a graph here, I can describe its key features around the roots.

At x = -4, the graph touches the x-axis and bounces back. Because the exponent 6 is even, the function doesn't change its sign here. If the graph is above the x-axis just before x = -4, it remains above the x-axis immediately after. Similarly, if it were somehow below (which it isn't in this particular function due to the other factor), it would remain below. The key is the bounce – a smooth turn without crossing.

On the other hand, at x = -7, the graph slices right through the x-axis. The exponent 5 is odd, so the function's sign flips. To the left of x = -7, f(x) is negative (the graph is below the x-axis). To the right of x = -7, f(x) becomes positive (the graph is above the x-axis). This clear transition from one side to the other is what defines a crossing.

If you were to sketch this graph, you'd see a curve approaching the x-axis at x = -7, passing through it, and continuing on the other side. At x = -4, you'd see the curve gently touching the x-axis and then turning back in the direction it came from. The graph's behavior perfectly illustrates the concept of multiplicity and its effect on x-axis crossings.

Examples and Further Exploration

Let's consider a few more examples to reinforce these concepts.

Example 1: g(x) = (x-2)^3 (x+1)^2

Here, the root x = 2 has a multiplicity of 3 (odd), so the graph crosses the x-axis at x = 2. The root x = -1 has a multiplicity of 2 (even), so the graph touches the x-axis at x = -1 but does not cross it.

Example 2: h(x) = (x+5)^4 (x-3)^7

In this case, the root x = -5 has a multiplicity of 4 (even), so the graph touches the x-axis at x = -5. The root x = 3 has a multiplicity of 7 (odd), so the graph crosses the x-axis at x = 3.

By analyzing the roots and their multiplicities, you can quickly determine the behavior of the graph near the x-axis. Remember, even multiplicities result in a touch (no crossing), while odd multiplicities result in a crossing.

Furthermore, you can explore more complex functions with multiple roots and varying multiplicities. Consider functions like k(x) = (x-1)^2 (x+2)^3 (x-4)^1. By examining each root and its multiplicity, you can sketch a rough graph of the function and understand its behavior. Remember to also consider the leading coefficient of the polynomial to determine the end behavior of the graph (whether it rises or falls as x approaches positive or negative infinity).

Conclusion

In conclusion, determining where a graph crosses the x-axis involves understanding the concepts of roots and their multiplicities. For the function f(x) = (x+4)^6 (x+7)^5, the graph crosses the x-axis at the root x = -7 because it has an odd multiplicity of 5. The root x = -4 has an even multiplicity of 6, so the graph touches the x-axis at that point but does not cross it. This knowledge allows us to analyze and understand the behavior of various functions and their graphs, providing valuable insights into their properties.

So, next time you encounter a function and need to determine its x-axis crossings, remember to identify the roots and their multiplicities. This simple yet powerful technique will help you visualize and understand the graph's behavior with ease. Keep exploring, keep learning, and you'll become a master of graphs in no time!