Let’s play a game. I’m going to draw two points on an x-y plane and give you a rule which you can use to generate new points. You can generate as many new points as you want using my rule. If you connect all the points, you get a function. Your goal is to guess the function. Ok? Let’s go.

Ok, I’ve drawn two points, $$a$$ and $$b$$. From these two points, you can generate a new third point $$c$$, which I’ve drawn in purple. The x coordinate of $$c$$ is the product of the x coordinates of $$a$$ and $$b$$. The y coordinate of $$c$$ is the sum of of the y coordinates of $$a$$ and $$b$$. That’s the rule.

To restate, here is my rule:

From any 2 points, $$a$$ and $$b$$, you can generate a 3rd point $$c$$ where $$c_x = a_x \cdot b_x$$ and $$c_y = a_y + b_y$$.

So… what’s the function? I’m looking for an answer of the form:

• your function is $$y = 2x^2$$
• or your function is $$y = xy + 5$$
• or something like that

Follow-up questions

If you think you have the answer, here are some follow-up questions:

• What if I picked two different starting points? How would the function change?
• Can I pick any two starting points? Or can I only pick certain points?
• Is my rule reversible? Can you, for example, use $$b$$ and $$c$$ to produce $$a$$? Why or why not? How can you convince yourself whether or not that’s allowed?