An owl swoops down from its perch in a tree at 30 feet to catch field mice below. It follows the function h= -6t^2-24t+30. Where t is the time and h is the height in feet. Which of the following best interprets the vertex?

To interpret the vertex of the quadratic function \( h(t) = -6t^2 - 24t + 30 \), we need to understand what the vertex represents in the context of this situation.

The function \( h(t) \) describes the height of the owl (in feet) at any given time \( t \) (in seconds) after it swoops down. The vertex of a quadratic function in the form \( h(t) = at^2 + bt + c \) represents the maximum (or minimum) point of the parabola, depending on the orientation.

Since the coefficient of \( t^2 \) (which is -6) is negative, the parabola opens downwards, which means the vertex will represent the maximum height that the owl reaches.

To find the vertex in terms of \( t \), we can use the formula for the \( t \)-coordinate of the vertex:

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

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

Plugging in the values:

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

Now, we'll substitute \( t = -2 \) back into the function \( h(t) \) to find the corresponding height \( h \):

\[ h(-2) = -6(-2)^2 - 24(-2) + 30 \] \[ = -6(4) + 48 + 30 \] \[ = -24 + 48 + 30 \] \[ = 54 \]

Thus, the vertex of the function is at the point \((-2, 54)\), meaning that at \( t = -2 \), the owl was at a maximum height of \( 54 \) feet.

However, none of the provided answer choices appear to match this interpretation directly.

Here's how each option relates to the context:

1. (6, 2) - This point does not correspond to the maximum height of the owl.
2. (2, 6) - Again, not relevant to height at any critical moment in the context.
3. (30, 0) - This indicates a point after the owl has stopped descending, which is not the vertex.
4. (0, 30) - Represents the initial height, not the vertex.

Since no option correctly interprets the vertex of the function \( h(t) \), it seems that there may have been a misunderstanding in the problem or the provided options do not include the actual vertex obtained from the calculations. In the context of interpreting the vertex, it indicates the maximum height the owl reaches relative to time, which is not reflected in the choices given.

Thanks for everything! it does not matter if the answer does not match. I want to learn so I will do it again on my own.

You're welcome! I'm glad I could help you understand the concept of interpreting the vertex in the context of the problem. It's great to hear that you're committed to learning, and I encourage you to continue practicing. If you ever have more questions or need clarification on any topic, feel free to reach out. Good luck with your studies!