Have you ever reflected deeply upon the mirror equation?

If you are looking at this blog-post, I will assume you have some sort of interest in curved mirrors.  Don't we all?  Who hasn't noticed how one's image on the inside of a shiny spoon looks different depending on how far away from one's eye one holds the spoon and whether one is looking at the inside or outside of the spoon? And talk about a simple experimental setup!  Physics is happening at nearly every breakfast table where cold cereal is eaten, tea is stirred, or sugar is sprinkled! 

This is an aspect of the behavior of light that fascinates nearly every curious mind at least some point in their life. It certainly fascinates me! Even though I know a few equations that help me understand the phenomenon to a deeper level than before, there is always more to learn! 

​I created the first version of this graph below during twenty minutes, a small part of one of my tutoring sessions with a local high school physics student.   It has already helped me gain a deeper understanding of the mirror equation, especially to visualize which circumstances lead to which results.
Take a look!  The best part about it is, it is interactive!   That's right!  You can drag the red dot marked "top of object" all over the graph to represent making an object bigger or smaller and closer or farther from a mirror, all at once!  

So what did you think of this graph?  If you have ever taken a physics course in which you encountered the mirror equation, you probably recognize this as a interactive version of the "ray diagram".   The straight lines trace the paths of the rays of light bouncing off of a spherically curved mirror from an object as viewed from the side.  Light travelling through a uniform material (or in a perfect a vacuum, the absence of all material) travels in a straight line.

The mirror equation describes where the "image" of an object will be if the object is placed in front of the curved mirror.  Although the mathematical equations describing this situation are not terribly complicated, they can be a little bit tricky to use!    The object can be positioned only in front of the mirror (some positive distance) but the radius of curvature of the mirror, and the distance in front of the mirror at which the image appears can both be negative. 
In the graph above, you can drag the slider to indicate where the top of the object appears in front of a concave spherical mirror.   
Q1. Can you find a situation with this mirror in which the magnification is very large and positive? Is the image real or virtual?
Q2.  Can you find a placement of the object in which the magnification is very large and negative?  Is the image real or virtual?
Q3.  Is there anything wrong with using this model to design an optical situation to achieve a large value of magnification? If so, describe as many of the problems as you can think of.

A1. By moving the object just to the right of the focal point you create a situation in which image is upright and virtual.  By moving the object very near to the focal point (but still to the right of it), you can see a situation where the magnification is very large and positive indeed. 
A2.  (IN THEORY, for what it is worth!) You can create a situation in which the magnification is very large but negative (in other words, inverted) if you move the object close to the focal point but just to the left of it. This image would be real. 
A3. If you over-did it, tried to make a mirror with massive magnification, in real life, I am certain you would not be able to get a very clear image.  One reason is because of spherical aberration.  Another reason is that you wouldn't expect this model to work once the image is taller than the mirror, right?  What happens ( or at least, doesn't happen) to the rays that were in theory supposed to pass through the focal point and reflect off the mirror to infinity in a direction parallel to the principal axis if the image height is greater than the height of the mirror?) Even if the mirror were adequately sized and shaped perfectly spherically, a spherical mirror does not really reflect rays passing through the focal point into a direction parallel to the principal axis.  It nearly does, only if the rays are nearly parallel with the principal axis.  And what about the image intensity?  The photons get spread out over a larger and larger area, so you can expect the image to get more and more faint too, right? Did you notice the rays in this mathematical model "reflect" off of the y-axis not the mirror itself? 

Like all models, the extent to which this model untethers itself from truth limits the degree to which we should believe it.  (Models this simple are only good for simple things, like analyzing what happens if the mirror doesn't change direction so much.)

You can see the virtual image itself (without leaving this web-page) if you move the object within 30 units of the y-axis.   (Or you can play around with the radius of curvature too and see how the equations are implemented if you click "edit graph on Desmos" in the bottom right corner of the graph).