# Thread: Transformation of functions and their graphs

1. ## Transformation of functions and their graphs

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

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

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

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

$\displaystyle H = 4T^2$

Thanks a whole bunch!

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

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

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

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

$\displaystyle H = 4T^2$

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

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

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

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

$\displaystyle \Rightarrow H = 4T^2$