Proportionality and Power Functions

Goals:

Explain proportionality and power functions in words, equations, and graphs
Describe the significance of a function being a power function
Determine if given data obeys a power rule, and if so, model it

What does it mean to be proportional/power function?

Definition: $ y $ is proportional to $ x $ if there is some constant $ k \neq 0 $ such that \[ y=kx \] $ k $ is called the proportionality constant

Definition: $ y $ is a power function if it is proportional to a constant power of $ x $ \[ y=kx^p \]

Proportionality is just a power function with power one. Note $ p $ can be negative, if so we say inversely proportional.

Examples:

Exchange rate from Dollars to Euros $ E=0.76D $. So Euros are proportional to Dollars with proportionality constant 0.76
Area of a circle $ A= \pi r^2 $: A is proportional to the square of its radius $ r $ with constant of proportionality $ pi $
Speed and travel time are inversely proportional because the faster you go, the shorter the time. As speed goes up, travel time goes down. $ r=\frac{D}{t}=Dt^{-1} $

Example:
Circulation time of a mammal (the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal.

Formula? $ T=kB^{1/4} $

Know: Elephant of body mass 5230 kg has a circulation time of 148 sec.

$ 148 = k 5230^{1/4} $ then $ k=17.40349 $ so $ T=17.4B^{1/4} $

So the circulation of a human with body mass 70 kg is $ T=17.4\cdot 70^{1/4}=50.33 $

Recognizing data from a power function

Cool Fact: Given a power function $ y=kx^p $, can transform into a line \[ \ln y=\ln kx^p=\ln k + \ln x^p =\ln k + p\ln x \]

Rename: $ \ln y =Y $, $ \ln k =b $, $ \ln x = X $. So $ Y=b+pX $

Given data points $ (x_1,y_1),\ (x_2,y_2) $, etc.
Form new data points $ (X_1,Y_1),\ (X_2,Y_2) $, etc., where $ X_1 = \ln x_1 $, $ Y_1 = \ln y_1 $, and so forth.
Plot the new points.
If the the plot looks linear, then the original data obey a power function with power $ p $ equal to the slope of the observed line, and with constant of proportionality $ k=\exp{b} $, where $ b $ is the observed vertical intercept.

Example: The minimum annual gross income, $ g $, in throusands of dollars, needed for a home loan of $100,000 at various interest rates, $ r $

r	g
8	31.447
9	34.484
10	37.611
11	40.814
12	44.084

Hypothesize $ g=kr^p $
Plot $ Y=\ln g $ vs. $ X= \ln r $

rates = 8:12
income = c(31.447, 34.484, 37.611, 40.814, 44.084)
logrates = log(rates)
logincome = log(income)

r	g	$ \ln r $	$ \ln g $
8	31.447	2.079	3.448
9	34.484	2.197	3.540
10	37.611	2.303	3.627
11	40.814	2.398	3.709
12	44.084	2.485	3.79

plotPoints(logincome ~ logrates)

plot of chunk unnamed-chunk-3

Find (approx.) best line of fit $ m=0.833 $ and $ b=1.712 $ (using fitModel in R)

plotPoints(logincome ~ logrates)
plotFun(0.833 * x + 1.712 ~ x, add = TRUE)

plot of chunk unnamed-chunk-5

-Then $ p=m=0.833 $ and $ k=e^b=e^{1.712}=5.54 $

So $ g=5.54 \cdot r^{0.833} $