Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] $f\left(x\right)=-x^3+x^2+2x$
a.b. c. d.

Verified step by step guidance

1. Identify the degree of the polynomial function \(f(x) = -x^3 + x^2 + 2x\). The degree is the highest power of \(x\), which is 3 in this case.
2. Determine the leading coefficient, which is the coefficient of the term with the highest degree. Here, the leading coefficient is \(-1\) (from \(-x^3\)).
3. Use the Leading Coefficient Test to analyze the end behavior: For an odd degree polynomial with a negative leading coefficient, as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
4. Summarize the end behavior: The graph falls to the right (because \(f(x) \to -\infty\) as \(x \to \infty\)) and rises to the left (because \(f(x) \to \infty\) as \(x \to -\infty\)).
5. Match this end behavior with the given graphs labeled (a) through (d) by selecting the graph that shows the function rising on the left and falling on the right.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps determine the end behavior of a polynomial function by examining the degree and the leading coefficient. For a polynomial f(x) = a_n x^n + ..., the sign of a_n and whether n is even or odd dictate how the graph behaves as x approaches ±∞.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It depends on the degree and leading coefficient: odd-degree polynomials with negative leading coefficients fall to the right and rise to the left, while even-degree polynomials rise or fall on both ends.

Matching Polynomial Functions to Graphs

Matching a polynomial to its graph involves using the end behavior, zeros, and general shape of the function. After determining end behavior via the Leading Coefficient Test, you compare these characteristics to the given graphs to identify the correct match.