# Thread: Transformation of functions and their graphs

1. ## Transformation of functions and their graphs

A ballet dancer jumps in the air. The height, $h(t)$, in feet, of the dancer at time $t$, in seconds since the start of the jump, is given by $^4$.

$h(t) = -16t^2 + 16Tt$

where $T$ is the total time in seconds that the ballet dancer is in the air.

Show that the time, $T$, that the dancer is in the air is related to $H$, the maximum height of the jump by the equation:

$H = 4T^2$

Thanks a whole bunch!

2. Hello hydride
Originally Posted by hydride
A ballet dancer jumps in the air. The height, $h(t)$, in feet, of the dancer at time $t$, in seconds since the start of the jump, is given by $^4$.

$h(t) = -16t^2 + 16Tt$

where $T$ is the total time in seconds that the ballet dancer is in the air.

Show that the time, $T$, that the dancer is in the air is related to $H$, the maximum height of the jump by the equation:

$H = 4T^2$

Thanks a whole bunch!
The velocity of the dancer at time $t$ is given by

$v = h'(t) = -32t+16T$

The dancer is at the maximum height when $v=0$; i.e. when $t = \tfrac12T$

So the maximum height H is given by $H=h(\tfrac12T)=-16(\tfrac12T)^2+8T^2$

$\Rightarrow H = 4T^2$