Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -5x^2 + 15x + 90$
answer
$\circ$ as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to \infty$
$\circ$ as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to -\infty$
$\circ$ as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to -\infty$
$\circ$ as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to \infty$

Explanation:

Step1: Identify degree and leading coefficient

The function $f(x) = -5x^2 + 15x + 90$ is a quadratic (degree 2, even) with leading coefficient $-5$ (negative).

Step2: Analyze end behavior for even degree

For even-degree polynomials, ends go in the same direction. A negative leading coefficient means both ends point downward.

• As $x \to \infty$, $x^2 \to \infty$, so $-5x^2 \to -\infty$, thus $y \to -\infty$.
• As $x \to -\infty$, $x^2 \to \infty$, so $-5x^2 \to -\infty$, thus $y \to -\infty$.

Answer:

as $x \to \infty, y \to -\infty$ and as $x \to -\infty, y \to -\infty$