Let's take your questions step by step.
Part A: Graph the Function
You can graph the function \( h(t) = 6t^2 - 24t + 30 \) using the Desmos graphing calculator. Here is how you can set it up:
- Go to the Desmos calculator website.
- Enter the equation:
h(t) = 6t^2 - 24t + 30. - Observe the parabola which opens upwards, as the coefficient of \( t^2 \) is positive.
Part B: Initial Height of the Hawk
To find the initial height of the hawk, we evaluate \( h(t) \) at \( t = 0 \):
\[ h(0) = 6(0)^2 - 24(0) + 30 = 30 \]
Therefore, the initial height of the hawk is 30 feet. The ordered pair for the initial height is:
Ordered Pair: \( (0, 30) \)
Part C: Lowest Height of the Hawk
To find the lowest height of the hawk, we need to find the vertex of the parabola represented by the function \( h(t) = 6t^2 - 24t + 30 \). The \( t \)-coordinate of the vertex can be found using the formula:
\[ t = -\frac{b}{2a} \]
Here, \( a = 6 \) and \( b = -24 \).
\[ t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2 \]
Now we can find the height at \( t = 2 \):
\[ h(2) = 6(2)^2 - 24(2) + 30 = 6(4) - 48 + 30 = 24 - 48 + 30 = 6 \]
Thus, the lowest height of the hawk is 6 feet. The ordered pair for this point is:
Ordered Pair: \( (2, 6) \)
Part D: Summary of the Lowest Point
The hawk will be at its lowest point at:
- Time: \( 2 \) seconds
- Height: \( 6 \) feet
So you can summarize:
- The hawk will be at its lowest point at time 2 seconds at a height of 6 feet.