Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=5x^3+7x^2-x+9$

Verified step by step guidance

1. Identify the degree of the polynomial function \(f(x) = 5x^3 + 7x^2 - x + 9\). 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 5.
3. Recall the Leading Coefficient Test rules for end behavior: For an odd degree polynomial, if the leading coefficient is positive, as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
4. Apply the test to this polynomial: Since the degree is 3 (odd) and the leading coefficient is 5 (positive), the graph will rise to the right and fall to the left.
5. Summarize the end behavior: As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity; as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients, combined using addition, subtraction, and multiplication. Understanding the general form and degree of a polynomial helps in analyzing its graph and behavior.

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine the end behavior of its graph. Specifically, it predicts how the function behaves as x approaches positive or negative infinity based on whether the degree is even or odd and the sign of the leading coefficient.

End Behavior of Functions

End behavior describes how the values of a function behave as the input x becomes very large or very small. For polynomials, this is determined by the highest-degree term, which dominates the function's growth or decline at the extremes of the x-axis.