Question

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?
(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 \( x \to \infty \).