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

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 \(x\). In this case, it is \$5x^4$.
2. Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, which is an even number.
3. Look at the leading coefficient, which is the coefficient of the leading term. Here, the leading coefficient is 5, a positive number.
4. Apply the Leading Coefficient Test: For an even degree polynomial with a positive leading coefficient, the end behavior is such that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
5. Summarize the end behavior based on the test: both ends of the graph will rise upwards toward 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. It states that the sign of the leading coefficient and whether the degree is even or odd 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 influences the shape and end behavior of the graph. For example, even-degree polynomials have similar end behaviors on both sides, while odd-degree polynomials have opposite end behaviors.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It helps predict whether the graph rises or falls on the far left and right sides, based on the leading term's degree and coefficient.