X-Axis Touch Points: F(x)=(x-5)^3(x+2)^2 Graph Analysis

Nov 7, 2025 by Admin 56 views

Hey guys! Let's dive into understanding how the graph of a polynomial function interacts with the x-axis, specifically focusing on the function f(x) = (x-5)^3(x+2)2. This is a super important concept in algebra and calculus, and we're going to break it down step by step. We will explore how the roots and their multiplicities affect whether the graph crosses or simply touches the x-axis. By the end of this discussion, you'll be able to easily identify the touch points of similar functions. Let's jump right in!

Understanding Roots and the X-Axis

First off, what are roots? In simple terms, the roots of a function f(x) are the values of x for which f(x) = 0. These are also the points where the graph of the function intersects or touches the x-axis. So, when we're talking about where a graph touches the x-axis, we're essentially looking for the roots of the function.

Now, let's consider our function, f(x) = (x-5)^3(x+2)2. To find the roots, we set f(x) equal to zero:

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

This equation is satisfied if either (x-5)^3 = 0 or (x+2)^2 = 0. Solving these gives us the roots x = 5 and x = -2. Great! We've found our roots. But there's more to the story than just the roots themselves.

The Significance of Multiplicity

This is where the concept of multiplicity comes into play. The multiplicity of a root refers to the number of times a particular factor appears in the factored form of the polynomial. It's like how many times a root is "repeated." In our function:

The factor (x-5) appears with an exponent of 3, so the root x = 5 has a multiplicity of 3.
The factor (x+2) appears with an exponent of 2, so the root x = -2 has a multiplicity of 2.

Multiplicity is super important because it tells us how the graph behaves at the x-axis. Specifically, it determines whether the graph crosses the x-axis or simply touches it and turns around. Let's dig deeper into how this works.

How Multiplicity Affects Graph Behavior

The multiplicity of a root has a direct impact on how the graph of the function interacts with the x-axis at that root. There are two main scenarios to consider:

Odd Multiplicity: If a root has an odd multiplicity (like 1, 3, 5, etc.), the graph crosses the x-axis at that point. Think of it as the graph passing right through the x-axis.
Even Multiplicity: If a root has an even multiplicity (like 2, 4, 6, etc.), the graph touches the x-axis at that point and then turns around. It doesn't cross over to the other side; it just kisses the x-axis and bounces back. This is often referred to as a turning point or a tangent point.

This difference in behavior is crucial for sketching polynomial graphs and understanding their properties. Now, let's apply this knowledge to our function f(x) = (x-5)^3(x+2)2.

Analyzing f(x) = (x-5)^3(x+2)2

We've already identified the roots of our function as x = 5 and x = -2. We also know their multiplicities:

x = 5 has a multiplicity of 3 (odd).
x = -2 has a multiplicity of 2 (even).

Using our rules about multiplicity, we can conclude the following:

At x = 5, the graph of f(x) will cross the x-axis because the multiplicity is odd.
At x = -2, the graph of f(x) will touch the x-axis and turn around because the multiplicity is even.

So, the key to answering our initial question—at which root does the graph touch the x-axis?—is the root with even multiplicity. In this case, that's x = -2.

Visualizing the Graph

To really solidify our understanding, let's think about what the graph of f(x) = (x-5)^3(x+2)2 might look like. We know a few key things:

The graph has roots at x = 5 and x = -2.
The graph touches the x-axis at x = -2 (even multiplicity).
The graph crosses the x-axis at x = 5 (odd multiplicity).

Additionally, we can consider the end behavior of the graph. The degree of the polynomial is the sum of the exponents, which is 3 + 2 = 5 (odd degree). The leading coefficient is positive (since there's no negative sign in front of the factored form). This means:

As x approaches positive infinity, f(x) also approaches positive infinity.
As x approaches negative infinity, f(x) approaches negative infinity.

Putting it all together, we can sketch a rough graph:

Start in the bottom-left quadrant (negative infinity).
The graph approaches the x-axis and touches it at x = -2, then turns back down.
The graph continues down, then turns and goes up, crossing the x-axis at x = 5.
The graph continues upwards towards positive infinity.

While we're not drawing an exact graph here, this mental picture helps us understand the behavior of the function and confirms our analysis of the roots and their multiplicities.

Real-World Applications and Further Exploration

Understanding how roots and multiplicities affect graph behavior isn't just a theoretical exercise. It has real-world applications in various fields, such as:

Engineering: Designing structures and systems that need to meet certain stability criteria.
Economics: Modeling market trends and predicting economic behavior.
Physics: Analyzing the motion of objects and the behavior of waves.

By mastering these fundamental concepts, you're building a strong foundation for tackling more complex problems in mathematics and beyond. If you're curious to explore further, consider looking into:

Graphing calculators and software: Tools like Desmos or GeoGebra can help you visualize functions and their graphs.
Calculus: The concepts of derivatives and tangent lines provide a deeper understanding of how graphs behave at turning points.
Polynomial interpolation: This technique uses polynomial functions to approximate data points, which has applications in data analysis and modeling.

Conclusion

So, to answer our initial question, the graph of f(x) = (x-5)^3(x+2)2 touches the x-axis at the root x = -2. This is because the root x = -2 has an even multiplicity (2), which means the graph bounces off the x-axis at that point rather than crossing it.

Understanding the relationship between roots, multiplicities, and graph behavior is a key skill in mathematics. It allows us to quickly analyze functions and visualize their graphs without needing to plot a bunch of points. Keep practicing with different functions, and you'll become a pro at identifying these touch points and understanding the behavior of polynomial graphs. Keep up the great work, guys, and happy graphing!