Hi Esther,
The monitor should wake up when the Mac wakes up, but sometimes there are synch issues where the monitor can't get the proper signal from your computer. After turning your monitor back on, try putting the computer to sleep and then waking it up again to see if the video signal synchs with the monitor. Also, iff the monitor is energy star compliant, you do not need to physically turn it off, it will go into low power mode.
good luck,
dave