Question

which statement is true about the end behavior of the graphed function?

as the x-values go to positive infinity, the functions values go to negative infinity.
as the x-values go to zero, the functions values go to positive infinity.
as the x-values go to negative infinity, the functions values are equal to zero.
as the x-values go to negative infinity, the functions values go to positive infinity.

Explanation:

Identify the target concept

We need to determine the correct statement describing the end behavior of the graphed function. End behavior describes what happens to the function's values \(y\) as the input values \(x\) become very large positive numbers (\(x \to \infty\)) or very large negative numbers (\(x \to -\infty\)).

Analyze the right side of the graph

Let's look at the behavior of the graph as we move to the far right along the x-axis:

• As \(x\) increases towards positive infinity (\(x \to \infty\)), the curve points sharply upwards.
• This means the function values \(y\) go to positive infinity (\(y \to \infty\)).

Analyze the left side of the graph

Let's look at the behavior of the graph as we move to the far left along the x-axis:

• As \(x\) decreases towards negative infinity (\(x \to -\infty\)), the curve points sharply upwards.
• This means the function values \(y\) go to positive infinity (\(y \to \infty\)).

Evaluate the given options

Let's compare our findings with the provided multiple-choice options:

• Option 1: "As the x-values go to positive infinity, the function's values go to negative infinity." This is incorrect because the graph goes up on the right.
• Option 2: "As the x-values go to zero, the function's values go to positive infinity." This is incorrect; at \(x = 0\), the function value is \(0\).
• Option 3: "As the x-values go to negative infinity, the function's values are equal to zero." This is incorrect; the graph goes up to infinity, not flat to zero.
• Option 4: "As the x-values go to negative infinity, the function's values go to positive infinity." This matches our analysis of the left side of the graph.

Answer:

• As the x-values go to positive infinity, the function's values go to negative infinity.
• As the x-values go to zero, the function's values go to positive infinity.
• As the x-values go to negative infinity, the function's values are equal to zero.
• As the x-values go to negative infinity, the function's values go to positive infinity. (Correct answer)