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

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

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

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

(2,6)

Answer 1

To interpret the vertex of the quadratic function \( h = 6t^2 - 24t + 30 \), we first need to find the vertex using the formula for the vertex of a parabola \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the standard form \( h = at^2 + bt + c \).

In your function, we can identify:

- \( a = 6 \)
- \( b = -24 \)

Plugging these values into the vertex formula gives us:

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

Now, we need to find the corresponding height \( h \) when \( t = 2 \):

\[
h = 6(2^2) - 24(2) + 30
\]
\[
h = 6(4) - 48 + 30
\]
\[
h = 24 - 48 + 30
\]
\[
h = 6
\]

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

Now, let's interpret this vertex:

- The \( t \)-coordinate (2) represents the time in seconds when the owl's height is at its maximum.
- The \( h \)-coordinate (6) represents the maximum height (in feet) that the owl reaches at that time.

So, the vertex \( (2, 6) \) indicates that at 2 seconds, the owl reaches a height of 6 feet. This means it drops down from a starting height of 30 feet to its lowest point, which is 6 feet.

Therefore, the best interpretation of the vertex is:

**(2, 6)**.