Quadratic Functions from Situations Quick Check

Question

5 of 55 of 5 Items
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

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

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

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

(30,0)
left parenthesis 30 comma 0 right parenthesis
Skip to navigation

Answer 1

To find the vertex of the quadratic function \( h = 6t^2 - 24t + 30 \), we can use the vertex formula for a quadratic in the form \( ax^2 + bx + c \). The \( t \)-coordinate of the vertex is given by:

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

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

Calculating the \( t \)-coordinate of the vertex:

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

Next, we need to find the \( h \)-coordinate by substituting \( t = 2 \) back into the equation:

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

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

So the best interpretation of the vertex in this context is:

\[ \boxed{(2, 6)} \]

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