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:
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:
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 = x2 and so we’ll start there.
The equation y = x2 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=x2 , 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 = x2. 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.
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.