The function you provided, \( y = -16t^2 + 30t + 2 \), represents the height \( y \) of the t-shirt at time \( t \) seconds after it is launched. This is a quadratic function in the form \( y = ax^2 + bx + c \), where \( a = -16 \), \( b = 30 \), and \( c = 2 \).
To find the vertex of the parabola represented by this function, we can use the formula for the vertex of a quadratic function given by:
\[ t = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{30}{2(-16)} = -\frac{30}{-32} = \frac{30}{32} = \frac{15}{16} \approx 0.9375 \text{ seconds} \]
Now, to find the height \( y \) at this time, substitute \( t = \frac{15}{16} \) back into the original equation:
\[ y = -16\left(\frac{15}{16}\right)^2 + 30\left(\frac{15}{16}\right) + 2 \] Calculating this step-by-step:
- \( \left(\frac{15}{16}\right)^2 = \frac{225}{256} \)
- \( -16\left(\frac{225}{256}\right) = -\frac{3600}{256} = -14.0625 \)
- \( 30\left(\frac{15}{16}\right) = \frac{450}{16} = 28.125 \)
- Adding everything: \[ y = -14.0625 + 28.125 + 2 \approx 16.0625 \]
So, the vertex of the parabola is approximately at:
\[ \left( \frac{15}{16}, 16.0625 \right) \]
Meaning:
The vertex of the parabola represents the maximum height reached by the t-shirt. In this case, the maximum height is approximately \( 16.0625 \) feet, and it occurs (\( t \)) about \( 0.9375 \) seconds after the t-shirt is launched. This means that at its peak (after a little less than 1 second), the t-shirt reaches a height of about \( 16.06 \) feet before it starts to fall back down to the ground. This is the highest point in its trajectory, which indicates when the cheerleaders can expect the t-shirt to be at its maximum elevation before descending into the crowd.