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

Question

, 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 $a{x}^2+bx+c$, 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.