Which graph has the same end behavior as the graph of $f(x)=-3 x^3-x^2+1$?

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natee805e natee805e

04-11-2026
Mathematics

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Which graph has the same end behavior as the graph of $f(x)=-3 x^3-x^2+1$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T12:55:52-05:00

The leading term of the given function $f(x) = -3x^3 - x^2 + 1$ is $-3x^3$, which has an odd degree ($n=3$) and a negative leading coefficient ($a_n=-3$).
The end behavior of any polynomial is governed by its leading term: for an odd degree and negative leading coefficient, the graph points toward $+\infty$ as $x \to -\infty$ and toward $-\infty$ as $x \to \infty$.
Any polynomial function with an odd degree and a negative leading coefficient will share this same end behavior.
The condition for a graph to have the same end behavior is: the function must be a polynomial of odd degree with a negative leading coefficient. $\boxed{\text{Odd degree, negative leading coefficient}}$

Explanation

Analysis of the Given Function

The end behavior of a polynomial function is primarily determined by its leading term, which is the term with the highest power of $x$. For the given polynomial $f(x) = -3x^3 - x^2 + 1$, the leading term is $-3x^3$. Here, the degree of the polynomial is $n = 3$, which is an odd number, and the leading coefficient is $a_n = -3$, which is a negative value. To determine the end behavior, we look at what happens as $x$ becomes very large in the positive or negative direction. As $x \to \infty$, the term $-3x^3$ becomes very large in the negative direction, so $f(x) \to -\infty$. Conversely, as $x \to -\infty$, the term $-3x^3$ becomes very large in the positive direction because $x^3$ is negative and the leading coefficient is negative, so $f(x) \to \infty$.

Determining Conditions for Same End Behavior

For any polynomial function $g(x) = b_m x^m + b_{m-1} x^{m-1} + \dots + b_0$, the end behavior is determined by the highest degree term $b_m x^m$. If the degree $m$ is odd, the ends of the graph will point in opposite directions. Specifically, if the leading coefficient $b_m$ is negative, the graph will go upward on the left (as $x \to -\infty$, $g(x) \to \infty$) and downward on the right (as $x \to \infty$, $g(x) \to -\infty$). This is exactly the same behavior as we found for $f(x) = -3x^3 - x^2 + 1$. Therefore, any polynomial function with an odd degree and a negative leading coefficient will exhibit the same end behavior as $f(x)$.

Conclusion and Summary of Characteristics

In conclusion, any graph that shares the same end behavior as $f(x) = -3x^3 - x^2 + 1$ must be represented by a polynomial function of odd degree $m$ with a negative leading coefficient $b_m$. Examples of such functions include any function of the form $g(x) = -a x^m + \dots$ where $m$ is an odd integer ($1, 3, 5, \dots$) and $a > 0$. The end behavior is characterized as: as $x \to \infty$, $y \to -\infty$, and as $x \to -\infty$, $y \to \infty$. Therefore, the graph of such a function will extend from the top-left quadrant to the bottom-right quadrant of the coordinate plane. The final characterization is that the function must be an odd-degree polynomial with a negative leading coefficient.

Examples

Understanding the end behavior of polynomials is useful in fields like economics or physics, where researchers model data using functions. For example, if an economist models the profit of a firm over time and finds that as time increases, the profit decreases significantly, they can use polynomials to predict future trends. Knowing whether a model will eventually grow to infinity or drop to negative infinity helps in long-term forecasting and strategic planning, ensuring that decisions are based on realistic expectations of a function's growth or decline.