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

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

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

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

(0,30)

Answer 1

To interpret the vertex of the function \( h = 6t^2 - 24t + 30 \), we can use the formula for the vertex of a quadratic function in the form \( h(t) = at^2 + bt + c \). The t-coordinate of the vertex can be found using the formula \( t = -\frac{b}{2a} \).

In this case:

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

Calculating the vertex's t-coordinate:

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

Next, we substitute \( t = 2 \) back into the function to find the h-coordinate (height):

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

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

This means that the interpretation of the vertex \( (2, 6) \) is as follows:

At \( t = 2 \) seconds, the height \( h \) of the owl is 6 feet. Since this is a quadratic function opening upward, the vertex represents the minimum point, meaning that after 2 seconds, the owl starts to rise from a height of 6 feet.

So, the best interpretation of the vertex is:

**(2, 6)**.