Question

question 7 | extrema and end behavior - practice... part b
b. use the given graph to describe the end behavior of the function
multiple-choice options (paraphrased):
a) as ( x \to -infty ), ( y \to -infty ) and as ( x \to infty ), ( y \to infty )
b) as ( x \to -infty ), ( y \to -infty ) and as ( x \to infty ), ( y \to -infty )
c) as ( x \to infty ), ( y \to infty ) and as ( x \to -infty ), ( y \to -infty )
d) as ( x \to -infty ), ( y \to -infty ) and as ( x \to infty ), ( y \to infty )
(graph of a polynomial function is shown)

Explanation:

To determine the end - behavior of the function from its graph:

Step 1: Analyze the left - hand end - behavior (as \(x

ightarrow-\infty\))
We observe the graph as \(x\) values become very large in the negative direction (move to the far left of the coordinate plane). From the given graph, as \(x
ightarrow-\infty\), the graph of the function is going downwards, which means \(y
ightarrow-\infty\).

Step 2: Analyze the right - hand end - behavior (as \(x

ightarrow\infty\))
We observe the graph as \(x\) values become very large in the positive direction (move to the far right of the coordinate plane). From the given graph, as \(x
ightarrow\infty\), the graph of the function is going upwards, which means \(y
ightarrow\infty\).

Looking at the options:

• Option A: As \(x

ightarrow-\infty,y
ightarrow\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\) is incorrect.

• Option B: As \(x

ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow-\infty\) is incorrect.

• Option C: As \(x

ightarrow\infty,y
ightarrow\infty\) and as \(x
ightarrow-\infty,y
ightarrow-\infty\) is incorrect. Wait, no, re - evaluating: Wait, the correct analysis from the graph (assuming the graph has a leading term with an odd degree and positive leading coefficient? Wait, no, the graph shown: when \(x\) goes to the left (negative infinity), the graph goes down ( \(y
ightarrow-\infty\)) and when \(x\) goes to the right (positive infinity), the graph goes up (\(y
ightarrow\infty\)). So the correct option should be the one where as \(x
ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\). If we assume the last option (let's say option D, since the user's options are a bit garbled but from the analysis) is the one with this behavior.

Answer:

The correct option (assuming the options are labeled and the one with as \(x
ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\)) is the one that matches this end - behavior. For example, if the options are:

A) As \(x
ightarrow-\infty,y
ightarrow\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\)

B) As \(x
ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow-\infty\)

C) As \(x
ightarrow\infty,y
ightarrow\infty\) and as \(x
ightarrow-\infty,y
ightarrow\infty\)

D) As \(x
ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\)

Then the answer is D. As \(x
ightarrow-\infty,y
ightarrow-\infty\) and as \(x
ightarrow\infty,y
ightarrow\infty\)