To interpret the vertex of the function given, we first need to recognize the function you provided:
\[ h(t) = 6t^2 - 24t + 30 \]
This is a quadratic function in the standard form \( h(t) = at^2 + bt + c \), where \( a = 6 \), \( b = -24 \), and \( c = 30 \).
The vertex of a quadratic function in the standard form is found using the formula:
\[ t = -\frac{b}{2a} \]
In this case:
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Calculate \( -\frac{b}{2a} \): \[ t = -\frac{-24}{2 \times 6} = \frac{24}{12} = 2 \]
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Now, substitute \( t = 2 \) back into the function to find \( h(2) \): \[ h(2) = 6(2^2) - 24(2) + 30 \] \[ = 6(4) - 48 + 30 \] \[ = 24 - 48 + 30 \] \[ = 6 \]
So, the vertex of the function is \( (2, 6) \).
Now, interpreting this vertex in the context of the problem:
- The \( x \)-coordinate (which corresponds to \( t \)) indicates that at \( t = 2 \) seconds, the height \( h \) is at its maximum value of 6 feet.
- Therefore, the vertex \( (2, 6) \) means that after 2 seconds, the swooping bird reaches its maximum height of 6 feet.
From your provided options, the best interpretation for the vertex \( (2, 6) \) then would be that the bird reaches its highest point of 6 feet at 2 seconds after it starts swooping down.
Thus, the answer is (2, 6).