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

a.(6,2)

b.(30,0)

c.(0,30)

d.(2,6)

Answer 1

To interpret the vertex of the given quadratic function \( h = 6t^2 - 24t + 30 \), we need to identify the vertex, which represents the maximum height of the owl as it swoops down.

The vertex can be found using the formula for the \( t \)-coordinate of the vertex, given by \( t = -\frac{b}{2a} \), where \( a \) is the coefficient of \( t^2 \) and \( b \) is the coefficient of \( t \).

Here, \( a = 6 \) and \( b = -24 \).

Plugging into the formula:

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

Next, we substitute \( t = 2 \) back into the original equation to find the \( h \)-coordinate of the vertex:

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

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

This means that at \( t = 2 \) seconds, the owl reaches its maximum height of 6 feet.

The best interpretation of the vertex is:

**d. (2, 6)**.