1-6 Modeling with functions
Modeling with Functions
After today’s lesson you should be able to
• Identify appropriate basic functions with
which to model real-world problems
• Produce specific functions to model data,
formulas, graphs, and verbal descriptions
Why develop models for functions?
• Think of two different ways to find the
circumference of a circle.
• Which method is more efficient?
Ex 1: Write the area of a circle as a
function of its
a) Radius r
b) Diameter d
c) Circumference C
• Ex 2: A square of size x inches is cut out of each
corner on an 8 in x 15 in piece of cardboard and the
sides are folded up to form an open topped box.
• Write the volume of the box as a function of x.
• Find the domain of x.
• Ex 3: A small satellite dish is packaged with a
cardboard cylinder for protection. Suppose the
parabolic dish has a 32 in. diameter and is 8 in.
deep. How tall must the 12 in diameter cylinder be
to fit in the middle of the dish and be flush with the
top of the dish?
• Ex 4: Water is stored in a conical tank with a faucet
at the bottom. The tank has depth 24 inches and
radius 9 in., and it is filled to the brim. If the faucet is
opened to allow the water to flow at a rate of 5 cubic
inches per second, what will the depth of the water
be after 2 minutes?
Functions from Data
Given a set of data points of the form (x,y), to construct a
formula that approximates y as a function of x:
1.
2.
3.
Make a scatter plot of the data points. The points do
not need to pass the vertical line test.
Determine from the shape of the plot whether the
points seem to follow the graph of a familiar type of
function (line, parabola, cubic, sine curve, etc.)
Transform a basic function of that type to fit the
points as closely as possible.
Regression Equations
• The effectiveness of a data-based model is highly
dependent on the number of data points and on
the way they were selected.
• Regression line – the line of best fit used to
describe a set of data.
• Correlation coefficient – measure of how well a
line models a set of data
• Denoted r
• Note: Used to describe linear data only
• Coefficient of determination – measure of how
well a non-linear equation models a set of data
• Denoted R2
Ex 5 The estimated number of U.S.
children that were home-schooled in
the years from 1992 to 1997 were:
a)
b)
c)
d)
Produce a scatter plot of the number
of children home-schooled in
thousands (y) as a function of years
since 1990 (x).
Find the linear regression equation.
Does the value of r2 suggest that the
linear model is appropriate?
Find the quadratic regression
equation. Does the value of R2
suggest that a quadratic model is
appropriate?
Use both curves to predict the
number of U.S. children that are
home-schooled in the year 2005.
How different are the estimates?
Year
1992
1993
1994
1995
1996
1997
Number
703,000
808,000
929,000
1,060,000
1,220,000
1,347,000