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

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

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

(6,2)

Answer 1

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

The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the equation.

In this case:
- \( a = 6 \)
- \( b = -24 \)

Calculating \( t \):

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

Next, we plug \( t = 2 \) back into the height function to find the corresponding height \( h \):

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

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

The vertex represents the maximum height that the owl reaches after 2 seconds.

Thus, the best interpretation of the vertex is:

**(2, 6)**.