In Exercises 15–18, 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).] <IMAGE>
$f\left(x\right)=-x^4+x^2$

Verified step by step guidance

1. Identify the leading term of the polynomial function. For the function \(f(x) = -x^4 + x^2\), the leading term is \(-x^4\) 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 \(-1\), a negative number.
4. Apply the Leading Coefficient Test: For an even degree polynomial, if the leading coefficient is negative, the ends of the graph both go down (toward negative infinity) as \(x\) approaches positive and negative infinity.
5. Use this end behavior to match the polynomial with the correct graph: the graph should show both ends falling downward, which corresponds to the behavior determined in the previous step.

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, if the leading coefficient is positive, both ends of the graph rise; if negative, both ends 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 general 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 on each side.

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 in matching the polynomial to its graph by predicting whether the graph rises or falls at the extremes.