Question

the graph of $y = f(x)$ is graphed below. what is the end behavior of $f(x)$?
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:

To determine the end behavior of \( f(x) \), we analyze the graph as \( x \) approaches \( \infty \) (right - hand end) and \( x \) approaches \( -\infty \) (left - hand end).

• As \( x \to \infty \): We look at the right - most part of the graph. The graph is rising upwards, which means as \( x \) gets larger and larger (approaches \( \infty \)), the value of \( y = f(x) \) also gets larger and larger, so \( y\to\infty \).
• As \( x \to -\infty \): We look at the left - most part of the graph. The graph is falling downwards, which means as \( x \) gets smaller and smaller (approaches \( -\infty \)), the value of \( y = f(x) \) also gets smaller and smaller, so \( y\to-\infty \).

Among the given options, the first option "as \( x \to \infty,y \to \infty \) and as \( x \to -\infty,y \to -\infty \)" matches our analysis of the end - behavior.

Answer:

as \( x \to \infty,y \to \infty \) and as \( x \to -\infty,y \to -\infty \) (the first option in the list of answer choices)