Question

The seat on a swing set sits 3 feet off the ground. The chains that are attached to the seat are 74 inches long. Suppose you were to swing: the path of the swing forms a parabola. Interpret the y-value of the vertex, in terms of the context.(1 point)
Responses

y≤74
y less-than-or-equal-to 74

y=3
y equals 3

y≤3
y less-than-or-equal-to 3

y=all real numbers

Answer 1

In this context, the vertex of the parabola represents the maximum height of the swing seat above the ground when it is at the highest point of its swing.

Given that the seat sits 3 feet off the ground (which is equivalent to 36 inches), the highest point that the seat can reach will be when it is at the point directly above the lowest point of the swing's arc (which is when the swing is at its peak).

The chains are 74 inches long, so the maximum y-value when the swing is at its highest point is:

\[
\text{Chain length} - \text{Height from the ground} = 74\, \text{inches} - 36\, \text{inches} = 38\, \text{inches}
\]

However, since the vertex pertaining to the height of the swing refers to the position of the swing seat above the ground when the swing is at rest (in terms of its motion), the actual maximum height the swing can achieve by following a smooth curved path and reflecting its range on the graph corresponds to the distance where the highest point perched before gravity pulls it back.

Based on all of this, the correct interpretation of the y-value of the vertex in terms of the context is:

**y=3**

This refers to the position of the swing seat 3 feet off the ground at its lowest point when the swing is at rest.