Linear Functions and Models

Lesson 2.1

Problems with Data

Real data recorded Experiment results Periodic transactions

Problems Data not always recorded accurately Actual data may not exactly fit theoretical

relationships In any case …

Possible to use linear (and other) functions to analyze and model the data

Fitting Functions to Data

Consider the data given by this example

Note the plot ofthe data points Close to being

in a straight line

TemperatureViscosity

(lbs*sec/in2)

160 28

170 26

180 24

190 21

200 16

210 13

220 11

230 9

Viscosity (lbs*sec/in2)

0

5

10

15

20

25

30

160 170 180 190 200 210 220 230 240

Finding a Line to Approximate the Data

Draw a line “by eye” Note slope, y-intercept

Statistical process (least squares method) Use a computer program

such as Excel Use your TI calculator

Chart Title

0

5

10

15

20

25

30

35

160 170 180 190 200 210 220 230 240

Graphs of Linear Functions

For the moment, consider the first option Given the graph with tic marks = 1

Determine Slope Y-intercept A formula for the function X-intercept (zero of the function)

Graphs of Linear Functions

Slope – use difference quotient

Y-intercept – observe Write in form

Zero (x-intercept) – what value of x gives a value of 0 for y?

change in y

change in x

ym

x

y m x b

Modeling with Linear Functions

Linear functions will model data when Physical phenomena and data changes at a constant

rate The constant rate is the slope of the function

Or the m in y = mx + b The initial value for the data/phenomena is the

y-intercept Or the b in y = mx + b

Modeling with Linear Functions

Ms Snarfblat's SS class is very popular. It started with 7 students and now, 18 months later has grown to 80 students. Assuming constant monthly growth rate, what is a modeling function? Determine the slope of the function Determine the y-intercept Write in the form of y = mx + b

Creating a Function from a Table

Determine slope by using

x y

3 7

4 8.5

5 10

6 11.5

change in y

change in x

ym

x

x y10 7 3

5 3 2

31.5

2

y

x

ym

y

Answer:

Creating a Function from a Table

Now we know slope m = 3/2 Use this and one of

the points to determiney-intercept, b

Substitute an orderedpair into y = (3/2)x + b

x y

3 7

4 8.5

5 10

6 11.5

310 5

220 3 5 2

5 2

5

2

b

b

b

b

3 5

:2 2

solution y x

Creating a Function from a Table

Double check results Substitute a different ordered pair into the

formula Should give a true statement

3 5:

2 2solution y x

x y

3 7

4 8.5

5 10

6 11.5

Piecewise Function

Function has different behavior for different portions of the domain

Greatest Integer Function

= the greatest integer less than or equal to x

Examples

Calculator – use the floor( ) function

( )f x x

6.7 6 3 3 2.5 3

Assignment

Lesson 2.1A Page 88 Exercises 1 – 65 EOO

15

Finding a Line to Approximate the Data

Draw a line “by eye” Note slope, y-intercept

Statistical process (least squares method) Use a computer program

such as Excel Use your TI calculator

Chart Title

0

5

10

15

20

25

30

35

160 170 180 190 200 210 220 230 240

16

You Try It

Consider table of ordered pairsshowing calories per minuteas a function of body weight

Enter data into data matrix ofcalculator APPS, Date/Matrix Editor, New,

Weight Calories

100 2.7

120 3.2

150 4.0

170 4.6

200 5.4

220 5.9

17

Using Regression On Calculator

Choose F5 for Calculations

Choose calculationtype (LinReg for this)

Specify columns where x and y values will come from

18

Using Regression On Calculator

It is possible to store the Regression EQuation to one of the Y= functions

19

Using Regression On Calculator

When all options areset, press ENTER andthe calculator comesup with an equation approximates your data

Note both the original x-y values and the function which

approximates the data

20

Using the Function Resulting function: Use function to find Calories

for 195 lbs. C(195) = 5.24

This is called extrapolation

Note: It is dangerous to extrapolate beyond the existing data Consider C(1500) or C(-100) in the context of the

problem The function gives a value but it is not valid

( ) 0.027 0.0169C x x Weight Calories

100 2.7

120 3.2

150 4.0

170 4.6

200 5.4

220 5.9

21

Interpolation

Use given data Determine

proportional“distances”

Weight Calories

100 2.7

120 3.2

150 4.0

170 4.6

195 ??

200 5.4

220 5.9

30 0.825 x

25

30 0.80.4167

4.6 0.4167 5.167

x

x

C

Note : This answer is different from the

extrapolation results

22

Interpolation vs. Extrapolation

Which is right? Interpolation

Between values with ratios Extrapolation

Uses modeling functions Remember do NOT go beyond limits of known data

( ) 0.027 0.0169C x x

23

Correlation Coefficient

A statistical measure of how well a modeling function fits the data

-1 ≤ corr ≤ +1

|corr| close to 1 high correlation

|corr| close to 0 low correlation

Note: high correlation does NOT imply cause and effect relationship

Assignment

Lesson 2.1B Page 94 Exercises 85 – 93 odd