Describe in words and symbols the end behavior of
f ( x )

Let's break down the end behavior of the function:

$f(x) = -5x^4$

• This is a polynomial function with an even degree (degree 4), and the leading coefficient is negative (-5).
• For even-degree polynomials, the behavior at both positive and negative infinity will be the same, but the sign of the leading coefficient determines whether the function increases or decreases.

Key points:

1. The function has a negative leading coefficient ($-5$).
2. The exponent is even ($x^4$), meaning that for very large positive or negative values of $x$, the term $x^4$ will dominate, and the sign of the coefficient will determine the direction.

End Behavior:

• As $x \to \infty$ (positive infinity), the $x^4$ term grows large, but because of the negative coefficient, the function $f(x)$ goes to negative infinity.
• As $x \to -\infty$ (negative infinity), the $x^4$ term still grows large (since raising a negative number to an even power results in a positive value), but again, because of the negative coefficient, $f(x)$ goes to negative infinity.

Thus, the correct end behavior is:

$\text{As } x \to \pm \infty, f(x) \to -\infty.$

Answer:

As x approaches positive or negative infinity, $f(x)$ decreases without bound:

$\text{As } x \to \pm \infty, f(x) \to -\infty$

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Would you like further clarification or a step-by-step breakdown of any specific part? Here are some related questions you could explore next:

1. How does the end behavior of polynomials with even degrees differ from those with odd degrees?
2. What is the significance of the leading coefficient in determining the end behavior of a polynomial?
3. How can we graph the function $f(x) = -5x^4$ and visualize its end behavior?
4. What happens to the end behavior of a polynomial if the leading coefficient is positive instead of negative?
5. How do the degree and coefficient of a polynomial function affect its general shape?

Tip: For polynomials, always check the degree and the leading coefficient first to quickly determine the end behavior of the function.