Which graph shows a polynomial function, f, such that f left-parenthesis x right-parenthesis right arrow infinity as x right arrow negative infinity and f left-parenthesis x right-parenthesis right arrow infinity as x right arrow infinity?

Question

(1 point)
Responses

A polynomial function is graphed on the coordinate plane. The curve decreases to the point left-parenthesis negative 1.87 comma negative 4.06 right-parenthesis, increases to the point left-parenthesis 0.54 comma 2.88 right-parenthesis, and then decreases infinitely.
Image with alt text: A polynomial function is graphed on the coordinate plane. The curve decreases to the point left-parenthesis negative 1.87 comma negative 4.06 right-parenthesis, increases to the point left-parenthesis 0.54 comma 2.88 right-parenthesis, and then decreases infinitely.

A polynomial function is graphed on the coordinate plane. The curve increases to the point left-parenthesis 0.91 comma 4.04 right-parenthesis and then decreases infinitely. There are arrows at both ends.
Image with alt text: A polynomial function is graphed on the coordinate plane. The curve increases to the point left-parenthesis 0.91 comma 4.04 right-parenthesis and then decreases infinitely. There are arrows at both ends.

A polynomial function is graphed on the coordinate plane. The curve increases to the point left-parenthesis negative 1 comma 4 right-parenthesis, decreases to the point left-parenthesis 1 comma 0 right-parenthesis, and then increases infinitely. There are arrows at both ends.
Image with alt text: A polynomial function is graphed on the coordinate plane. The curve increases to the point left-parenthesis negative 1 comma 4 right-parenthesis, decreases to the point left-parenthesis 1 comma 0 right-parenthesis, and then increases infinitely. There are arrows at both ends.

A polynomial function is graphed on the coordinate plane. The curve decreases to the point left-parenthesis negative 1.08 comma negative 1.05 right-parenthesis and then increases infinitely.
Image with alt text: A polynomial function is graphed on the coordinate plane. The curve decreases to the point left-parenthesis negative 1.08 comma negative 1.05 right-parenthesis and then increases infinitely.

Answer 1

To determine which graph represents a polynomial function \( f \) such that \( f(x) \to \infty \) as \( x \to -\infty \) and \( f(x) \to \infty \) as \( x \to \infty \), we need to understand the behavior of polynomial functions based on their degrees and leading coefficients.

For a polynomial function to approach \( \infty \) in both directions:

1. Degree of the polynomial: The degree must be even (e.g., 2, 4, etc.).
2. Leading coefficient: The leading coefficient must be positive.

From the responses you provided, we can analyze each option:

1. First graph: The function decreases to a minimum point and then increases again but does not tend to infinity at both ends. This graph doesn't meet the requirement.

2. Second graph: The function increases to a maximum and then decreases infinitely. This indicates it does not meet the requirement either.

3. Third graph: This graph shows the function increasing on both sides to infinity, as it has arrows on both ends. This is indeed a polynomial function that behaves as desired.

4. Fourth graph: The function decreases to a point and then increases infinitely, which means it does not go to infinity as \( x \) approaches \( -\infty\). Thus, it does not meet the requirement.

Based on this analysis, the graph you are looking for is the third graph, as it fits the requirement of \( f(x) \to \infty \) as \( x \to -\infty \) and \( f(x) \to \infty \) as \( x \to \infty \).