Power Models & Proportionality
Power models of the form y=axb or y∝xb, including direct proportion y=kxn, inverse proportion y=xnk, and fitting a power model from data using power regression.
Direct Proportion
Directly proportional quantities are constant multiples of each other. In the context of modelling, we typically say, "y varies directly with xn," which means y=kxn for some constant k. This can be denoted y∝xn.
If y is directly proportional to xn, then x=0⟺y=0.
If y is directly proportional to xn, then if x increases (or decreases) by a factor of c, y increases (or decreases) by a factor of cn.
Inverse proportion
If y varies inversely with xn, then y=xnk.
If y is inversely proportional to xn (y∝xn1), then the y-axis is an asymptote of the graph of y=f(x).
Fitting a power model
Proportionality relations can be used to build models called power models, which have the form
which is equivalent to saying y∝xb.
Power models can be found from given data using your calculator's power regression feature.
Nice work completing Power Models & Proportionality , here's a quick recap of what we covered:
Skills covered
Exercises checked off