The equation given is \( h(t) = -6t^2 - 24t + 20 \). This is a quadratic function in the standard form \( h(t) = at^2 + bt + c \), where \( a = -6 \), \( b = -24 \), and \( c = 20 \).
To find the vertex of this parabolic function, we can use the formula for the \( t \)-coordinate of the vertex, which is given by:
\[ t = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{-24}{2 \cdot -6} = \frac{24}{-12} = -2 \]
Next, we find the \( h \) value at \( t = -2 \) by substituting back into the original equation:
\[ h(-2) = -6(-2)^2 - 24(-2) + 20 \] \[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 24 \]
Thus, the vertex of the function is \((-2, 24)\).
However, in the provided responses, the closest and mistakenly given option seems to be:
- \((-2, 44)\) instead of the actual \((-2, 24)\).
None of the provided answers reflect the correct vertex. So, the correct vertex is not available in the options given.
If you must select the closest one from the provided answers, recognize that:
- \( (-2, 44) \) is the only one with the correct \( t \)-coordinate, although the corresponding \( h \)-value is incorrect.
Overall, the correct answer based solely on calculated results is that the vertex is \( (-2, 24) \), but you might indicate the response \( (-2, 44) \) if it's necessary to pick from the options provided.
1 answer