Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=11x^3-6x^2+x+3$

Verified step by step guidance

1. Identify the leading term of the polynomial function. The leading term is the term with the highest power of \(x\), which in this case is \$11x^{3}$.
2. Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 3, an odd number.
3. Look at the leading coefficient, which is the coefficient of the leading term. Here, the leading coefficient is 11, a positive number.
4. Apply the Leading Coefficient Test for end behavior: For an odd degree polynomial with a positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
5. Summarize the end behavior: The graph falls to the left (as \(x\) approaches negative infinity) and rises to the right (as \(x\) approaches positive infinity).

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine its end behavior. The sign and parity (odd or even) of the leading term dictate how the graph behaves as x approaches positive or negative infinity.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It indicates the general shape of the graph and influences the number of turning points and the end behavior of the polynomial function.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined by the leading term and helps predict whether the graph rises or falls on either end.