Linear relationships can describe a great many things, but they’re not perfect. For example, suppose we want to predict the Michigan’s grey wolf population next year using the population numbers from this year. A linear relationship might look like this:

[picture]

The problem with this is that it predicts unlimited growth: the more wolves there are this year, the more there will be next year. And that’s just not true! It’s tru-ish for small populations, but less true for large populations, because Michigan can only support so many wolves. We should expect a relationship that looks more like this one:

[picture]

In fact, a relationship like this can give us an extremely good description of Michigan’s wolf population between 1988 and today. Stop by my office and I’ll show you!

## The Model Quadratic

We should care about nonlinear models because many interesting and useful phenomena in life are nonlinear.

The simplest nonlinear thing is just *y* = *x*^{2} and so we’ll start there.

The equation *y* = *x*^{2} is an example of a *quadratic*. We can think of *y* as being the area of a square whose sidelengths are *x. *This is why an exponent of two is called “squaring.” Area interpretations will occasionally help us understand things about quadratics.

We’re eventually going to see that *all* quadratics are more or less the same as *y*=*x*^{2} , just sort of stretched out, flipped over, and moved around. So we need to understand this guy as much as possible.

- Create a graph of
*y*=*x*^{2}. What does the graph look like? What striking features does this graph have? - What does the table of values look like? What kinds of patterns can you find in this table?
- Find ways to articulate connections between the graph, the equation, and the table. Explain your connections to someone who’s willing to listen. For example: a friend; a tutor; Marc Boucher; or me.

[diagram]

You should find robust answers to these questions – not for my sake but for your own understanding.

Once you feel like you’ve got a good grip on this relationship, let’s move on to translations.