Question

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

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

Explanation:

To determine the end - behavior of the function \(y = f(x)\) from its graph, we analyze the direction of the graph as \(x\) approaches positive infinity (\(x
ightarrow\infty\)) and negative infinity (\(x
ightarrow-\infty\)).

1. **As \(x

ightarrow\infty\)**:

• We look at the right - hand end of the graph. From the given graph, as \(x\) gets larger and larger (moves towards the right along the \(x\) - axis), the graph of the function is going downwards, which means that \(y = f(x)\) approaches \(-\infty\), so as \(x

ightarrow\infty\), \(y
ightarrow-\infty\).

1. **As \(x

ightarrow-\infty\)**:

• We look at the left - hand end of the graph. As \(x\) gets smaller and smaller (moves towards the left along the \(x\) - axis), the graph of the function is also going downwards, which means that \(y = f(x)\) approaches \(-\infty\), so as \(x

ightarrow-\infty\), \(y
ightarrow-\infty\).

Answer:

As \(x
ightarrow\infty,y
ightarrow-\infty\) and as \(x
ightarrow-\infty,y
ightarrow-\infty\) (the first option among the given choices)