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<\/p>\n<\/p>\n
Linear relationships can describe a great many things, but they’re not perfect.\u00a0 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:<\/p>\n
[picture]<\/p>\n
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.\u00a0 We should expect a relationship that looks more like this one:<\/p>\n
[picture]<\/p>\n
In fact, a relationship like this can give us an extremely good description of Michigan’s wolf population between 1988 and today.\u00a0 Stop by my office and I’ll show you!<\/p>\n
We should care about nonlinear models because many interesting and useful phenomena in life are nonlinear.<\/p>\n
The simplest nonlinear thing is just y<\/em> =\u00a0x<\/em>2<\/sup> and so we’ll start there.<\/p>\n The equation y<\/em> =\u00a0x<\/em>2<\/sup> is an example of a quadratic<\/em>.\u00a0 We can think of\u00a0y<\/em> as being the area of a square whose sidelengths are x.\u00a0 <\/em>This is why an exponent of two is called “squaring.”\u00a0 Area interpretations will occasionally help us understand things about quadratics.<\/p>\n We’re eventually going to see that all<\/em> quadratics are more or less the same as\u00a0y<\/em>=x<\/em>2<\/sup> , just sort of stretched out, flipped over, and moved around.\u00a0 So we need to understand this guy as much as possible.<\/p>\n [diagram]<\/p>\n You should find robust answers to these questions – not for my sake but for your own understanding.<\/p>\n\n