Letβs 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.