Question

A hawk swoops down from its perch 30 feet in a tree to catch field mice. It follows the function h(t)=6t^2−24t+30

, where t
𝑡
is time in seconds and h
ℎ
is height in feet.

Part A: Graph this function. Link to Desmos calculator

Part B: What is the initial height of the hawk? Write your answer as an ordered pair

Part C: At which point is the hawk at it's lowest height? Write your answer as an ordered pair

Part D: The hawk will be at it's lowest point at time seconds at a height of feet



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Answer 1

Let’s break this down step by step.

### Part A: Graphing the Function
You can graph the function \( h(t) = 6t^2 - 24t + 30 \) using a graphing calculator like Desmos. To do this, simply enter the function into the graphing interface.

### Part B: Initial Height of the Hawk
The initial height of the hawk occurs when \( t = 0 \). To find this, substitute \( t = 0 \) into the function:

\[
h(0) = 6(0)^2 - 24(0) + 30 = 30
\]

Thus, the initial height in ordered pair form is:

\[
(0, 30)
\]

### Part C: Lowest Height of the Hawk
To find the lowest point of the parabola defined by the quadratic function, 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:

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

In this case, \( a = 6 \) and \( b = -24 \):

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

Now plug this value back into the function to find the height:

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

Thus, the lowest height occurs at the ordered pair:

\[
(2, 6)
\]

### Part D: Final Statement
The hawk will be at its lowest point at time \( 2 \) seconds at a height of \( 6 \) feet.

### Summary of Answers
- **Part B**: Initial height: \((0, 30)\)
- **Part C**: Lowest height: \((2, 6)\)
- **Part D**: The hawk will be at its lowest point at time \( 2 \) seconds at a height of \( 6 \) feet.