5. Quadratic Functions in Context(1 point)

Question

A hawk swoops down from its perch 30 feet in a tree to catch field mice. It follows the function h(t)=6t2−24t+30
, where t
is time in seconds and 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

Answer 1

Let's break down each part of the problem step by step.

Part A: Graph the Function

To graph the function \( h(t) = 6t^2 - 24t + 30 \), you can use a graphing tool like Desmos. Here’s how you can do it:

1. Open Desmos Graphing Calculator.
2. Enter the function \( h(t) = 6t^2 - 24t + 30 \).
3. You should see a parabola opening upwards, representing the trajectory of the hawk's height over time.

Part B: Initial Height of the Hawk

The initial height of the hawk can be found by evaluating \( h(t) \) at \( t = 0 \):

\[ h(0) = 6(0)^2 - 24(0) + 30 = 30 \text{ feet} \]

So, the ordered pair representing the initial height is:

\[ (0, 30) \]

Part C: Lowest Height of the Hawk

To find the lowest point of the parabola, we can find the vertex of the quadratic function, which is given by the formula \( t = -\frac{b}{2a} \). Here, \( a = 6 \) and \( b = -24 \).

Calculating \( t \):

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

Now, substitute this value back into the function to find the lowest height:

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

The ordered pair for the lowest height is:

\[ (2, 6) \]

Part D: Summary

The hawk will be at its lowest point at time \( 2 \) seconds at a height of \( 6 \) feet.

So, to summarize:

• Part B: \( (0, 30) \)
• Part C: \( (2, 6) \)
• Part D: The hawk will be at its lowest point at time 2 seconds at a height of 6 feet.