To solve the problem presented, we need to analyze the function \( h(t) = 6t^2 - 24t + 30 \) step by step.
Part A: Graph this function
To graph the function, you can use the Desmos calculator. You would input the equation \( h(t) = 6t^2 - 24t + 30 \) into the graphing tool, and it will display the parabola that represents the height of the hawk over time.
Part B: What is the initial height of the hawk?
The initial height occurs when \( t = 0 \). To find the initial height, we substitute \( t = 0 \) into the function:
\[ h(0) = 6(0)^2 - 24(0) + 30 = 30 \]
So, the initial height of the hawk is 30 feet. As an ordered pair, the initial height is:
\[ (0, 30) \]
Part C: At which point is the hawk at its lowest height?
To find the lowest point (the vertex of the parabola), we can use the vertex formula for a quadratic function, which is given by \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the function \( h(t) = at^2 + bt + c \).
Here, \( a = 6 \) and \( b = -24 \).
Calculating \( t \):
\[ t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2 \]
Now, substitute \( t = 2 \) back into the height function to determine the height at this time:
\[ h(2) = 6(2)^2 - 24(2) + 30 = 6(4) - 48 + 30 = 24 - 48 + 30 = 6 \]
Thus, the lowest point of the hawk's flight occurs at:
\[ (2, 6) \]
Part D: The hawk will be at its lowest point at time __________ seconds at a height of __________ feet.
As we calculated in Part C, the hawk will be at its lowest point at:
- Time: \( 2 \) seconds
- Height: \( 6 \) feet
So, you can fill in the blanks as:
The hawk will be at its lowest point at time 2 seconds at a height of 6 feet.
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