An owl swoops down from its perch in a tree at 30 feet to catch field mice. It follows the function h=6t^2 −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
(30,0)
left parenthesis 30 comma 0 right parenthesis
(6,2)
left parenthesis 6 comma 2 right parenthesis
(0,30)
To find the vertex of the parabola represented by the quadratic function \( h(t) = 6t^2 - 24t + 30 \), we need to find the time \( t \) at which the height \( h \) is at a minimum for this downward-opening parabola.
The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) is found at the point \( (h, k) \), where \( h = -\frac{b}{2a} \) and \( k \) is the value of the function when \( t = h \).
In the given function \( h(t) = 6t^2 - 24t + 30 \), we have:
\( a = 6 \) and \( b = -24 \).
Let's calculate \( h \), the time at which the vertex occurs:
\( h = -\frac{-24}{2 \cdot 6} \)
\( h = \frac{24}{12} \)
\( h = 2 \) seconds.
Now we will find the height \( k \) at \( t = 2 \) seconds by substituting it into the function:
\( k = h(2) = 6(2)^2 - 24(2) + 30 \)
\( k = 6(4) - 48 + 30 \)
\( k = 24 - 48 + 30 \)
\( k = 6 \) feet.
Therefore, the vertex of the parabola is \( (2, 6) \), which is the point where the owl is at the minimum height above the ground during its swoop. This is the first response given in the list:
(2,6)
left parenthesis 2 comma 6 right parenthesis