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).]
$f\left(x\right)=x^6-6x^4+9x^2$

Verified step by step guidance

1. Identify the leading term of the polynomial function. For the given function \(f(x) = x^6 - 6x^4 + 9x^2\), the leading term is \(x^6\) 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 6, an even number.
3. Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is 1 (positive).
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. Use this end behavior information to match the polynomial function with the graph that shows both ends of the graph rising upwards.

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 even-degree polynomials with a positive leading coefficient, both ends of the graph rise; if negative, both fall. For odd-degree polynomials, the ends go in opposite directions depending on the sign of the leading coefficient.

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. In this problem, the polynomial degree is 6, an even number, which affects how the graph behaves as x approaches positive or negative infinity.

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 match the polynomial to its graph by predicting whether the graph rises or falls on each end.