Use the image to answer the question.

Question

A horizontal S-shaped curve is plotted on a coordinate plane with the x-axis ranging from negative 4 to 2 in increments of 0.5 and the y-axis ranging from negative 2 to 6 in increments of 0.5.

Which of the following most accurately identifies the relative extrema of the polynomial function h(x)
?

(1 point)
Responses

There is a relative minimum at (0,−2)
and a relative maximum at (4,0)
.
There is a relative minimum at left parenthesis 0 comma negative 2 right parenthesis and a relative maximum at left parenthesis 4 comma 0 right parenthesis .

There is a relative maximum at (−2,0)
and a relative minimum at (0,4)
.
There is a relative maximum at left parenthesis negative 2 comma 0 right parenthesis and a relative minimum at left parenthesis 0 comma 4 right parenthesis .

There is a relative minimum at (−2,0)
and a relative maximum at (0,4)
.
There is a relative minimum at left parenthesis negative 2 comma 0 right parenthesis and a relative maximum at left parenthesis 0 comma 4 right parenthesis .

There is a relative maximum at (0,−2)
and a relative minimum at (4,0)
.

Answer 1

Based on the description of the S-shaped curve, we can identify key features such as relative minima and maxima.

1. A relative maximum occurs at a peak point in the curve where the function changes direction from increasing to decreasing.
2. A relative minimum occurs at a trough point in the curve where the function changes direction from decreasing to increasing.

Given the options provided:

• For the first option, relative minimum at (0, -2) and relative maximum at (4, 0) doesn't align with typical S-shaped curves, as a max at (4, 0) would imply the function is increasing towards (4, 0) and then is decreasing, which conflicts with the S-shape.
• For the second and third options regarding maxima and minima at points such as (−2, 0) and (0, 4), these also seem inconsistent with the S-shape.

Without the specific visual of the graph, the most common understanding of an S-shaped curve is that it has:

• A relative maximum in the center (higher up) where it turns from increasing to decreasing.
• A relative minimum at the bottom end where it turns from decreasing to increasing.

Thus the one that best matches typical behavior for an S-shape is:

There is a relative maximum at (−2, 0) and a relative minimum at (0, 4).

This aligns with the general expected behavior of polynomial functions modeled in an S-curve format.