Question

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. Plot points to graph the equation. Which ordered pair represents the vertex?

Answer Options:

(1, 0)
(-5, 0)
(44, -2)
(-2, 44)

Answer 1

To find the vertex of the quadratic function \( h(t) = -6t^2 - 24t + 20 \) and to graph it, we can use the vertex formula. The vertex of a parabola given by the equation \( h(t) = at^2 + bt + c \) can be found at \( t = -\frac{b}{2a} \).

Here, we have:
- \( a = -6 \)
- \( b = -24 \)

Calculating the t-coordinate of the vertex:

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

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

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

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

However, your provided answer options do not include this vertex, so let's analyze them carefully:

1. \( (1, 0) \)
2. \( (-5, 0) \)
3. \( (44, -2) \)
4. \( (-2, 44) \)

None of these points represent the vertex of the quadratic function we calculated. As a result, it appears there may be an error with the answer options provided. The correct vertex calculated is \( (-2, 24) \).

To further assist you, if you would like to know more about plotting this quadratic function or have further questions, please let me know!