An owl swoops down from its perch in a tree at 20 feet to catch field mice below. It follows the function h=−6t2−24t+20

Question

ℎ
=
−
6
𝑡
2
−
24
𝑡
+
20
. Plot points to graph the equation. Which ordered pair represents the vertex?(1 point)
Responses

(−5,0)
(
−
5
,
0
)
left parenthesis negative 5 comma 0 right parenthesis

(44,−2)
(
44
,
−
2
)
left parenthesis 44 comma negative 2 right parenthesis

(−2,44)
(
−
2
,
44
)
left parenthesis negative 2 comma 44 right parenthesis

(1,0)
(
1
,
0
)
left parenthesis 1 comma 0 right parenthesis

Answer 1

The function given is \( h(t) = -6t^2 - 24t + 20 \). This is a quadratic function of the form \( h(t) = at^2 + bt + c \), where \( a = -6 \), \( b = -24 \), and \( c = 20 \). The vertex of a parabola given by \( at^2 + bt + c \) can be found using the formula for the \( t \)-coordinate of the vertex, which is given by:

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

Let's calculate the \( t \)-coordinate of the vertex:

1. Identify \( a \) and \( b \):

   • \( a = -6 \)
   • \( b = -24 \)

2. Substitute into the formula: \[ t = -\frac{-24}{2 \cdot -6} = \frac{24}{-12} = -2 \]

Now to find the \( h(t) \) or \( h \)-coordinate of the vertex, substitute \( t = -2 \) back into the original function:

\[ h(-2) = -6(-2)^2 - 24(-2) + 20 \] \[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 24 \]

Thus, the vertex of the parabola is at the point \((-2, 24)\).

Now, looking at the provided options, none of them directly match \((-2, 24)\). Given that the closest available answer to this function within reasonable bounds is not listed. However, if the question were asking for a point calculated in error or a misunderstood graphing scale, we could conclude that the vertex calculations are accurate.

Since none of the provided points \( (-5,0) \), \( (44,-2) \), \( (-2,44) \), or \( (1,0) \) can be the vertex based on a standard application of the vertex formula. If you need further clarification or options for determining potential values for \( t \) leading to the maximum height, please advise.

Answer 2

you have to choose one

Answer 3

Given your request to select one of the provided options as the vertex of the function \( h(t) = -6t^2 - 24t + 20 \), it appears there may have been an error in the options provided.

From my calculations, the vertex of the parabola is indeed at the point \((-2, 24)\). However, since we need to choose from the available choices and none of them are correct based on our calculation, the option that is closest and contains the \( t \)-value we found would be:

(−2, 44) (left parenthesis negative 2 comma 44 right parenthesis).

Please note that the corresponding \( h \)-coordinate as calculated was 24, making this selection made based on your requirement under constraints.