The horizontal blue line through the middle of the lens is called the optical axis. as the lamp is moved above or below the axis, keeping the horizontal distance to the center of the lens constant, how does the horizontal distance from the lens to the image change?

As the object moves parallel to principle axis the image of the object will continuously moves away from the principal axis so the horizontal distance will increase

Explanation:

As the object is moving towards the lens then in that case first the image will form between the focus and double the distance of focus.

So here the image is diminished

now when the object is at large distance from from lens then its image will form at focus and it is of very small size

Now as the object comes closer to the lens till the object distance is 2F the image size will increase but smaller then object size

Now when object is between F and 2F then the image size will be more than object size and it will increase to infinite magnification

So overall we can say that as the object moves parallel to principle axis the image of the object will continuously moves away from the principal axis so the horizontal distance will increase