Section 2.2 – Power Functions w/ Modeling
Learning Outcomes:
-
Find a power regression that models given data (calculator)
Understand and apply the terms direct and inverse variation
Key Vocab Terms:
Constant of variation (proportion): the variable k represents the constant of variation value for either
f ( x)  k x a for x  0 if f(x) is a direct variation or y x a  k if f(x) is an inverse variation, EVERY power
function will have a defined constant of variation
Direct Variation: a power function that when written in the form f ( x)  k x a , the power is positive
y
The equation can also be written in the form: a  k . Graphically the function goes through the origin.
x
Inverse Variation: a power function that when written in the form f ( x)  k x a , the power is negative
The equation can also be written in the form: y x a  k . Graphically the function does NOT go through the
origin.
These four power function models involve output-from-input relationships that can be expressed in the language
of variation and proportion:
• The circumference of a circle varies directly as its radius.
• The area enclosed by a circle is directly proportional to the square of its radius.
• The force of gravity acting on an object is inversely proportional to the square of the distance from the object
to the center of the Earth.
• Boyle’s Law states that the volume of an enclosed gas (at a constant temperature) varies inversely as the
applied pressure.
The power function formulas with positive powers are statements of direct variation and power function
formulas with negative powers are statements of inverse variation.
Unless the word inversely is included in a variation statement, the variation is assumed to be direct.
Examples to work through:
1. From empirical evidence and the laws of physics it has been found that the period of time T for the full
swing of a pendulum varies as the square root of the pendulum’s length l, provided that the swing is
small relative to the length of the pendulum. Write an equation that represents the relationship between
the time for a full swing and the pendulum’s length.
2.
Write an equation for each of the relationships:
a. The weight of an object, w, is directly proportional to the mass of the object, m, given the
constant of acceleration due to gravity, c.
b. The refractive index of a medium, n, varies inversely with the velocity of light, v, in the medium
given a constant velocity of light, c, in free space.
3.
A dripping faucet wastes a cup of water if it drips for three minutes.
a. What type of relationship is there between the amount of time that has passed and the amount of
water that has been wasted?
b. How long will it take for the faucet to waste 4.5 cups of water?
4. The number of buckets of paint n needed to paint a fence varies directly with the total area of the fence
and inversely with the amount of paint p in a bucket. It takes three 1-gallon buckets of paint to paint 72
square feet of fence. How many 1-gallon buckets will be needed to paint 90 square feet of fence?
Just because a function only has one term does not mean that the function is a monomial function. In
order to be a monomial function, it must be a polynomial function.
Entering data into a calculator and looking at the relationship of the data as a scatter plot allows us to
determine if there seems to be a correlation between the independent and dependent values. Up to now
the data we have looked at seemed to be linear and so we have calculated the linear regression to get an
equation that best models the relationship. Now we are going to be looking at data that may be best
modeled as a power function.
Enter the following data giving the average distances a planet is from the sun and the number of days it takes
that planet to orbit the sun. (Gm stands for gigameters = 1 million kilometers)
Look at a scatter plot of the data.
Does there seem to be a correlation between the distance and period of the orbit?
What type of function do you think would best model the correlation?
Find the regression equation that best models the correlation. ________________________________________
What is the coefficient of determination (r2, how well the model fids the data) in this situation and so what can
you conclude about how well the regression represents the data?
What is the power of the function? ______________
Using the regression, how many days would you estimate it to take Pluto which is 7400 Gm away from the sun
(yes, I know it is not considered a planet now) to orbit the sun? _______________