Use one of the end behavior diagrams below, to describe the end behavior of the graph of each polynomial function.

$\text{ƒ}\left(x\right)=-4x^3+3x^2-1$

Verified step by step guidance

1. Identify the leading term of the polynomial function. For the function \(f(x) = -4x^3 + 3x^2 - 1\), the leading term is \(-4x^3\) 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 3, an odd number.
3. Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is \(-4\), a negative number.
4. Use the degree and leading coefficient to describe the end behavior: For an odd degree polynomial with a negative leading coefficient, as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
5. Summarize the end behavior in an end behavior diagram or notation: as \(x \to -\infty\), \(f(x) \to \infty\); as \(x \to \infty\), \(f(x) \to -\infty\).

Key Concepts

Polynomial Functions and Their Degrees

A polynomial function is an expression consisting of variables raised to whole-number exponents with coefficients. The degree of the polynomial is the highest exponent of the variable, which largely determines the shape and end behavior of its graph. For example, in ƒ(x) = -4x^3 + 3x^2 - 1, the degree is 3.

Leading Coefficient and Its Effect on End Behavior

The leading coefficient is the coefficient of the term with the highest degree. It influences the direction the graph heads as x approaches positive or negative infinity. A negative leading coefficient, like -4 in the example, typically causes the graph to fall on the right end and rise on the left end for odd-degree polynomials.

End Behavior Diagrams

End behavior diagrams visually represent how the graph of a polynomial behaves as x approaches positive or negative infinity. They use arrows or symbols to show whether the function values rise or fall at the ends. These diagrams help summarize the long-term trends of polynomial graphs without plotting every point.