To find the vertex of the quadratic function \( h = 6t^2 - 24t + 30 \), we can use the vertex formula for a parabola in the form \( h = at^2 + bt + c \), where \( a = 6 \), \( b = -24 \), and \( c = 30 \).
The time at which the vertex occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
Substituting in the values of \( a \) and \( b \):
\[ t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2 \]
Next, we can find the height \( h \) at this time by substituting \( t = 2 \) back into the height equation:
\[ h = 6(2)^2 - 24(2) + 30 \]
Calculating this step-by-step:
\[ h = 6(4) - 48 + 30 = 24 - 48 + 30 = 6 \]
Thus, the vertex of the function is \( (2, 6) \), meaning that:
- The owl reaches its maximum height of 6 feet above the ground at 2 seconds after it begins to swoop down.
Interpretation of the vertex: The vertex represents the point in time when the owl is 6 feet above the ground, which is the lowest height it reaches in its swoop (since the parabola opens upwards). Therefore, the best interpretation of the vertex in this context is:
At 2 seconds, the owl is 6 feet above the ground, which is the lowest point in its flight path before it continues to descend further.
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