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 Radius r Diameter d 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 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: 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 Ex 5 The estimated number of U.S. children that were home-schooled in the years from 1992 to 1997 were: 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 Number 1992 1993 1994 1995 1996 1997 703,000 808,000 929,000 1,060,000 1,220,000 1,347,000