Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=-5x^4+7x^2-x+9$

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1. Identify the leading term of the polynomial function. For the given function \(f(x) = -5x^4 + 7x^2 - x + 9\), the leading term is \(-5x^4\) because it has the highest power of \(x\).
2. Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, an even number.
3. Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is \(-5\), a negative number.
4. Apply the Leading Coefficient Test rules: For an even degree polynomial, if the leading coefficient is positive, both ends of the graph go up; if negative, both ends go down. Since the leading coefficient is negative, both ends of the graph will fall as \(x\) approaches \(\pm \infty\).
5. Summarize the end behavior: As \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to non-negative integer powers, 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, the sign and parity (even or odd) of the leading term dictate whether the graph rises or falls as x approaches positive or negative infinity.

End Behavior of Graphs

End behavior describes how the values of a function behave as the input x approaches very large positive or negative values. For polynomials, this is primarily influenced by the leading term, indicating whether the graph rises or falls on each end.