Question

Marco is given 2 equations that appear very different and asked to graph them using Desmos. He notices that even though the equations appear very different, the graphs overlap perfectly.Explain why this is possible?

Answer 1

See below for a couple of ideas:

Explanation:

There are a couple of answers here.

It's the same equation but in different form

If I graph `y=x` and then I play around with the equation, not changing the domain or range, I can have the same basic relation but with a different look:

graph{x}

`2(y-3)=2(x-3)`

graph{2(y-3)-2(x-3)=0}

The graph is different but the grapher doesn't show it

One way this can show up is with a small hole or discontinuity. For instance, if we take that same graph of `y=x` and put a hole in it at `x=1`, the graph won't show it:

`y=(x)((x-1)/(x-1))`

graph{x((x-1)/(x-1))}

First let's acknowledge that there is a hole at `x=1` - the denominator is undefined there. So why is there no hole?

The reason is that the hole is only at 2.00000....00000. The points right next to it, 1.9999...9999 and 2.00000....00001 are valid. The discontinuity is infinitely small and so the grapher won't show it.