Analyzing Polynomial Graphs: A Detailed Explanation

Nov 17, 2025 by Alex Johnson 52 views

Understanding the behavior of polynomial functions is a fundamental skill in mathematics. The ability to analyze a polynomial function's graph allows us to quickly determine key characteristics like its roots (where it crosses the x-axis) and its overall shape. In this article, we'll delve into the process of analyzing the graph of a polynomial function, using the example function $f(x) = x^4 + x^3 - 2x^2$ . We'll break down the function, explain how to find its roots, and then relate those roots to the graph's behavior. We will explore how to identify whether the graph crosses or touches the x-axis at specific points, making sure you gain a solid grasp of this important concept. This will help you confidently solve problems related to polynomial functions and their graphical representations.

Factoring the Polynomial Function

The first step in analyzing the graph of a polynomial function like $f(x) = x^4 + x^3 - 2x^2$ is to factor it. Factoring allows us to identify the roots of the function easily. Roots are the x-values where the function equals zero, and these points are where the graph intersects the x-axis. Let's factor the given function step by step:

Identify the Greatest Common Factor (GCF): In this case, the GCF of all terms is $x^2$ . We can factor out $x^2$ from each term: $f(x) = x^2(x^2 + x - 2)$ .
Factor the Quadratic Expression: Now, we need to factor the quadratic expression inside the parentheses, $x^2 + x - 2$ . We are looking for two numbers that multiply to -2 and add up to 1 (the coefficient of the x term). These numbers are 2 and -1. Therefore, we can factor the quadratic expression as $(x + 2)(x - 1)$ .
Complete Factored Form: Combining all factors, the fully factored form of the function is: $f(x) = x^2(x + 2)(x - 1)$ .

This factored form is crucial because it directly reveals the roots of the polynomial function. Each factor that includes x will contribute to the roots of the equation. Understanding this process is key to linking the algebraic form of the function to its graphical representation. The factors give us the x-intercepts, where the graph crosses or touches the x-axis, and understanding these factors will help to accurately describe the behavior of the polynomial's graph. Correctly factoring a polynomial is, therefore, the cornerstone for accurate analysis.

Determining the Roots and Their Significance

Once we have the factored form of the polynomial function, $f(x) = x^2(x + 2)(x - 1)$ , we can easily determine its roots. The roots are the values of x that make the function equal to zero. To find the roots, we set each factor equal to zero and solve for x.

From the factor $x^2$ : $x^2 = 0$ . Solving for x gives us $x = 0$ . This root has a multiplicity of 2, which means it appears twice in the factored form. This affects how the graph behaves at this x-intercept. When a root has an even multiplicity, the graph touches the x-axis but does not cross it. Therefore, at x = 0, the graph touches the x-axis.
From the factor $(x + 2)$ : $x + 2 = 0$ . Solving for x gives us $x = -2$ . This root has a multiplicity of 1, meaning it appears once. When a root has an odd multiplicity (like 1), the graph crosses the x-axis at that point. Thus, the graph crosses the x-axis at x = -2.
From the factor $(x - 1)$ : $x - 1 = 0$ . Solving for x gives us $x = 1$ . This root also has a multiplicity of 1, so the graph crosses the x-axis at x = 1.

In summary, the roots of the function are x = -2, x = 1, and x = 0. The root at x = 0 has a multiplicity of 2, while the roots at x = -2 and x = 1 each have a multiplicity of 1. These roots and their multiplicities are key to understanding the graph's behavior. The distinction between roots with odd and even multiplicities is crucial as it determines whether the graph crosses or touches the x-axis, providing insights into the polynomial's overall shape. Remember this relationship, and you'll find it far easier to match the equations with the respective graphs.

Matching Roots to Graph Behavior

Now that we've found the roots and understood their multiplicities, we can describe how the graph of the polynomial function $f(x) = x^4 + x^3 - 2x^2$ behaves at those points. The roots are the x-intercepts, and the multiplicity of each root determines whether the graph crosses or touches the x-axis.

At x = -2: The graph crosses the x-axis. This is because the factor $(x + 2)$ has a multiplicity of 1. At this point, the function's value changes sign, moving from positive to negative or vice versa as it crosses the x-axis.
At x = 1: The graph also crosses the x-axis. The factor $(x - 1)$ has a multiplicity of 1, leading to a sign change as the graph passes through this x-intercept.
At x = 0: The graph touches the x-axis. The factor $x^2$ has a multiplicity of 2. Because the multiplicity is even, the graph touches the x-axis at x = 0 but does not cross it. The function's value does not change sign at this point; it comes down, touches the x-axis, and then turns back up.

This behavior is characteristic of polynomial functions. The degree of the polynomial (in this case, 4) impacts the overall shape, but the roots and their multiplicities define the graph's specific interactions with the x-axis. Thus, knowing the roots and their multiplicities is crucial to predicting the graph's shape without actually plotting the points. This is an essential skill when working with polynomials.

Evaluating the Given Statements

Now, let's look at the multiple-choice options provided, keeping in mind what we've discovered about our polynomial's graph. The question asks us to identify the statement that correctly describes the graph of $f(x) = x^4 + x^3 - 2x^2$ .

Option A: