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Power Models & Proportionality
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Modeling with functions of the form axn, and the concept of direct and inverse variation
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Direct Proportion
SL AI 2.5
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
SL AI 2.5
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
SL AI 2.5
Proportionality relations can be used to build models called power models, which have the form
y=a⋅xb
which is equivalent to saying y∝xb.
Power models can be found from given data using your calculator's power regression feature.
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