As mentioned the waveform applied to the zener will determine the output waveform. With out knowing one you cannot know the other....
However you can plot the characteristics of a zener on a VI chart, where the X-axis is Voltage and the Y-axis is current.
Since all diodes have a Zener breakdon voltage the VI chart of a diode will look similar from one divice to another. Where the only differences of note will be the point at which breakdown ocurrs. Zeners are doped differently so as to improve the breakdown characteristics and consistency between divices.
If you plot the current vs voltage characterisitic of a zener diode you will find that the graph for a zener resembles a regular diode.
As voltage is applied in the forward direction (to left on X axis) from zero volts the Zener will not initially conduct so the I (current will be zero) But as the forward bias is increased from zero it must oppose and overcome the forward juction potential of (0.7 or 0.63Volts).
As the forward bias (voltage is increased) to about 0.7V the forward junction voltage is overcome and the diode will beging to strongly conduct (down on the Y Axis). A very slight incresase in forward voltage will cause a dramatic nearly verticle rise in current. Since we do not have a series resistor included in this plot the forward current is quite large .
No further rise in voltage can be plotted as the diode is in full conduction.
Now for reverse Bias applied to the diode the PI curve for a zener is exaclt like that of a regular diode EXCEPT a zener will have a well defined and repeatable breakdown voltage with a sharper knee on the curve.
On reverse bias, voltage is applied in the opposite direction and the diode will not conduct as the reverse biase aids the junction voltage strenghtening it. So as voltage is increased from zero the current through the diode stays at zero. so the plot line stays horizontal. As voltage rises and approaches the breakdown voltage some current will manage to flow through the diode causing a slight rise in current. (This is the knee of the curve.) Add alittle more voltage and zener breakdown occurrs and the current will rise sharply. This will send the plot line in a verticle direction. THe diode is in conduction with a high current flow and no further (practicle) increase in voltage can be plotted.
EDIT:
See data sheet :
THere is a VI curve on the second page where forward voltage is plotted to the right