To interpret the coordinates of the vertex in the context of the given quadratic equation [tex]\( h = -16t^2 + 128t \)[/tex]:
1. The quadratic equation [tex]\( h = -16t^2 + 128t \)[/tex] models the height [tex]\( h \)[/tex] of the rocket as a function of time [tex]\( t \)[/tex].
2. The vertex of this quadratic equation represents the maximum height of the rocket, as the parabola opens downward (the coefficient of [tex]\( t^2 \)[/tex] is negative).
To find the coordinates of the vertex, we use the following steps:
- The [tex]\( t \)[/tex]-coordinate of the vertex of the parabola given by [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
For our equation [tex]\( h = -16t^2 + 128t \)[/tex], the coefficients are [tex]\( a = -16 \)[/tex] and [tex]\( b = 128 \)[/tex]. Plugging in these values:
[tex]\[
t = \frac{-128}{2 \cdot (-16)} = \frac{128}{32} = 4
\][/tex]
Therefore, the [tex]\( t \)[/tex]-coordinate of the vertex is 4 seconds.
- To find the corresponding [tex]\( h \)[/tex]-coordinate (the height at this moment), we substitute [tex]\( t = 4 \)[/tex] back into the original equation:
[tex]\[
h = -16(4)^2 + 128(4)
\][/tex]
[tex]\[
h = -16 \cdot 16 + 128 \cdot 4
\][/tex]
[tex]\[
h = -256 + 512 = 256
\][/tex]
Therefore, the [tex]\( h \)[/tex]-coordinate of the vertex is 256 feet.
Interpreting these coordinates in context:
- The [tex]\( t \)[/tex]-coordinate (or x-coordinate) of the vertex is 4 and represents the time in seconds at which the rocket reaches its maximum height.
- The [tex]\( h \)[/tex]-coordinate (or y-coordinate) of the vertex is 256 and represents the maximum height in feet that the rocket reaches.
In conclusion:
- The [tex]\( t \)[/tex]-coordinate (or [tex]\( x \)[/tex]-coordinate) of the vertex is [tex]\( 4 \)[/tex] and represents the time (in seconds) at which the rocket reaches its maximum height.
- The [tex]\( h \)[/tex]-coordinate (or [tex]\( y \)[/tex]-coordinate) of the vertex is [tex]\( 256 \)[/tex] and represents the maximum height (in feet) that the rocket reaches.
