Question

An owl swoops down from its perch in a tree at 30 feet to catch field mice. It follows the function h=6t2 −24t+30
h
=
6
t
2

−
24
t
+
30
, where t is the time in seconds and h is the height in feet. Which of the following best interprets the vertex?(1 point)
Responses

(2,6)
(
2
,
6
)
left parenthesis 2 comma 6 right parenthesis

(6,2)
(
6
,
2
)
left parenthesis 6 comma 2 right parenthesis

(30,0)
(
30
,
0
)
left parenthesis 30 comma 0 right parenthesis

(0,30)

Answer 1

To interpret the vertex of the quadratic function \( h = 6t^2 - 24t + 30 \), we first need to find the coordinates of the vertex.

The vertex of a quadratic function in the form \( at^2 + bt + c \) can be calculated using the formula for the time \( t \) at which the maximum or minimum occurs:

\[
t = -\frac{b}{2a}
\]

In this case, \( a = 6 \) and \( b = -24 \). Plugging in these values:

\[
t = -\frac{-24}{2 \times 6} = \frac{24}{12} = 2
\]

Next, we find the height \( h \) at \( t = 2 \) by substituting \( t \) back into the original equation:

\[
h = 6(2)^2 - 24(2) + 30
\]

Calculating this step-by-step:

\[
h = 6(4) - 48 + 30
\]
\[
h = 24 - 48 + 30
\]
\[
h = 6
\]

Thus, the vertex is at the point \( (2, 6) \).

Interpreting this in the context of the owl's height:

- **The x-coordinate (2)** represents the time in seconds when the owl reaches its highest point.
- **The y-coordinate (6)** represents this maximum height in feet.

Therefore, the best interpretation of the vertex \( (2, 6) \) is that at 2 seconds, the owl is at a maximum height of 6 feet.

The correct answer is:

**(2, 6)**.