To find the vertex of the quadratic function \( h = -6t^2 - 24t + 20 \), we can use the vertex formula for a quadratic equation \( at^2 + bt + c \). The \( t \)-coordinate of the vertex can be found using the formula \( t = -\frac{b}{2a} \).
In this function:
- \( a = -6 \)
- \( b = -24 \)
Substituting the values into the formula:
\[ t = -\frac{-24}{2 \times -6} = \frac{24}{-12} = -2 \]
Next, we will find the corresponding \( h \)-coordinate by substituting \( t = -2 \) back into the equation:
\[ h = -6(-2)^2 - 24(-2) + 20 \]
Calculating this step by step:
\[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 24 + 20 = 44 \]
Thus, the vertex of the parabola is at the point (-2, 44).
Now we can match this with the options given:
- A: (1, 0)
- B: (-5, 0)
- C: (-2, 44)
- D: (44, -2)
The correct ordered pair that represents the vertex is:
C: (-2, 44).