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
, 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)
left parenthesis 0 comma 30 right parenthesis

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

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

(30,0)

Answer 1

To interpret the vertex of the quadratic function given by the equation \( h = 6t^2 - 24t + 30 \), we need to find the time \( t \) at which the height \( h \) reaches its maximum or minimum point.

The vertex of a quadratic function in the form \( h = at^2 + bt + c \) can be found using the formula:

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

In your equation, \( a = 6 \) and \( b = -24 \). Plugging in the values:

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

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

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

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

In the context of the problem, this means that at \( t = 2 \) seconds, the owl is at a height of \( 6 \) feet, which represents either a minimum height if the parabola opens upwards (which it does, since \( a = 6 > 0 \)).

Therefore, the best interpretation of the vertex is:

**(2, 6)**.