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

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

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

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

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

Next, we can find the corresponding height \( h \) at this time by substituting \( t = 2 \) back into the height function:

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

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

### Interpretation:
- The x-coordinate (or t-coordinate) of the vertex is **2 seconds**, which represents the time at which the rocket reaches its maximum height.
- The y-coordinate (or h-coordinate) of the vertex is **64 feet**, which represents the maximum height of the rocket.

In summary:

- 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 of the rocket**.