Question

A rocket is launched in the air. Its height in feet is given by h, equals, minus, 16, t, squared, plus, 64, th=−16t
2
+64t where tt represents the time in seconds after launch. Interpret the coordinates of the vertex in context.
Answer
Attempt 1 out of 2

The x-coordinate (or t-coordinate) of the vertex is
and represents

.
The y-coordinate (or h-coordinate) of the vertex is
and represents

.

Answer 1

To find the vertex of the quadratic function \( h(t) = -16t^2 + 64t \), we can use the vertex formula. The x-coordinate (or t-coordinate) of the vertex for a quadratic function in the form \( ax^2 + bx + c \) is given by:

\[
t = -\frac{b}{2a}
\]

In this case, \( a = -16 \) and \( b = 64 \). Plugging in these values:

\[
t = -\frac{64}{2 \cdot -16} = -\frac{64}{-32} = 2
\]

So, the x-coordinate of the vertex is \( t = 2 \) seconds.

Next, we'll find the corresponding height (y-coordinate or h-coordinate) at \( t = 2 \):

\[
h(2) = -16(2^2) + 64(2)
\]
\[
= -16(4) + 128
\]
\[
= -64 + 128
\]
\[
= 64
\]

Therefore, the y-coordinate of the vertex is \( h = 64 \) feet.

### Interpretation:

- The **x-coordinate (or t-coordinate) of the vertex is** 2 seconds and represents the time at which the rocket reaches its maximum height.

- The **y-coordinate (or h-coordinate) of the vertex is** 64 feet and represents the maximum height that the rocket reaches during its flight.

In summary:
- The vertex of the function is at \( (2, 64) \), which indicates that the rocket reaches its maximum height of 64 feet at 2 seconds after launch.