Animating z surface math functions

If you don't want to code the animation, you can use Animation Nodes to do that.

Animation Nodes is an addon which can be found here: https://github.com/JacquesLucke/animation_nodes/releases

Here is a setting that allows to influence a Z curve depending on the position of an empty object (but that can be almost anything including the current frame).

The principle is in two parts: a main loop and its execution for each iteration of the loop.

The main loop:

It starts from 2 parameters, a subdivided plane (1) and an empty (2).

From the plane, the vertices are extracted so that they are given as input for the execution function (7), and the other element (edges, polygons) are reported to be combine to a new mesh (4) and to create a new 'target' object (5).

From the empty, we take its location (6) which goes as input for the execution function (7).

Note that the plane can be either an existing object in the scene or generated by AN using a grid.

The execution function

We can see the defined inputs in (1).

The first input 'object vertices' is given by the calling node and corresponds to each vertex of the plane (the loop iterates giving 1 vertex per iteration, in this case).

The second input 'center' corresponds to the vector which is the location of the empty.

These two vectors are decomposed in (2) and (3) to get the coordinates we want.

In (4) additional parameters (they should be set outside of the loop, if needed).

In (5) an 'expression node' which takes all input parameters and allows to write the curve formula.

In (6) we recompose the vector, keeping X and Y from the input and Z from the formula.

(6) is given as output in (7) so that we can have it at the "upper level" (the loop 'caller').

Animate using shape keys and drivers.

Via UI add a math surface shape, then another with changed equation parameters. Do not change the number of x and y divisions to ensure the new has the same vertex count as the former. Repeat.

When done select all the variations and join as shapes. Animate the shapekey values via keyframe and drivers.

A script to do it.

Example script using z function surface addon. Default equation 1 - (a * x**2 + y**2) for a = [0..9]. Create a shape for each using bpy.ops.mesh.primitive_z_function_surface(...) (the operator from the addon) passing the substituted a to the operator as the equation property. (eg for a=3 equation="1 - (3 * x**2 + y**2)" .)

The separate objects are joined using bpy.ops.object.join_shapes()

import bpy
context = bpy.context 
# 10 shapes over 101 frames

scene = context.scene
vl = context.view_layer
shapes = 10
obs = []

for i in range(shapes):
    bpy.ops.mesh.primitive_z_function_surface(
                    equation="1 - (%d * x**2 + y**2)" % i,
                    div_x=16,
                    div_y=16,
                    size_x=2,
                    size_y=2)

    ob = context.object
    ob.name = "a=%d" % i

    obs.append(ob)

shape = obs.pop(0)
vl.objects.active = shape

for ob in obs:
    ob.select_set(True)

bpy.ops.object.join_shapes()
for ob in obs:
    bpy.data.objects.remove(ob)

# add a discrete driver animation (10 frames of each)
keys = shape.data.shape_keys.key_blocks

for i in range(1, shapes):
    shape = keys.get("a=%d" % i)
    d = shape.driver_add("value")
    d.driver.expression = "frame // 10 == %d" % i

Note the animation uses drivers. Make sure autorun is enabled

Make sure the Add Mesh : Extra Objects addon is enabled too

NURBS

As a side note, a NURBS surface gives a reasonable approximation of a paraboloid

Image below depicts moving the bottom rectangle of the nurbs points in local z direction.

The default nurbs surface has been scaled and translated to overlay the z-surface.

Adding Hooks on the surface control points could be a investigated. Related Parametizing parabolic nurbs surface