1

Functions and Linear ModelsChapter1

• Functions: Numerical, Algebraic and Graphical

• Linear Functions

• Linear Models

• Linear Regression

Lecture 1

2

Functions

A real-valued function f assigns to each real number x in a specified set of numbers (the domain of f ), a unique real number f (x), read “f of x.”

* The natural domain of f is the largest set of numbers for which f makes sense.

NOTE: It is not f times x

3

Numerically Specified Function

Ex.

tt 00 11 22 33 44

VV((tt)) 2.22.2 3.553.55 4.94.9 6.256.25 7.67.6

The data represents the velocity of an object V, in feet/sec, after t seconds have elapsed.

* Note: at 2 seconds the object is going 4.9 ft/sec.

Given numerical values for the function evaluated at certain values of the independent variable.

4

Algebraically Defined FunctionAlgebraically Defined Function

2( ) 3 2f x x is a function.

( )f x h

Ex.

2(5) 3(5) 2 77f

23 2x h

2 23 6 3 2x xh h

A function represented by a formula.

5

Mathematical ModelingRepresenting a situation in mathematical terms.

Ex. The monthly payment, M, necessary to repay a home loan of P dollars, at a rate of r % per year (compounded monthly), for t years, can be found using

12

12

112 12

1 112

t

t

r rP

Mr

6

Common Types of Algebraic Functions

Linear

Quadratic

Polynomial

( )f x mx b

2( )f x ax bx c

11 0( ) ...n n

n nf x a x a x a

(a, b, m, and each ai constant)

(a not 0)

(an not 0)

7

Common Types of Algebraic Functions

Exponential (A, b constant, b >0)

Rational (P, Q polynomials)

( ) xf x Ab

( )( )

( )

P xf x

Q x

8

Piecewise Function

Several formulas to define a single function.

Ex. 2

32 5.5 if 2( )

13.8 2.5 if 2

x xf x

x x

(1) 32 5.5(1)f 2(3) 13.8 2.5(4)f

= 26.5

= 53.8

Use when x values are less than or equal to 2

Use when x values are greater than 2

Notice

9

Graphically Specified FunctionGraphically Specified FunctionThe graph of a function is the set of all points (x, f (x)) such that x is in the domain of f .

Given the graph of y = f (x), find f (1).

f (1) = 2(1, 2)

10

Graph of a FunctionGraph of a FunctionVertical Line Test: The graph of a function can be crossed at most once by any vertical line.

Function Not a Function

It is crossed more than once.

11

Sketching a Piecewise Function

2

2 if 2 1( )

+1 if 1 2

x xf x

x x

Sketch the portion of the formula on its domain

12

Linear FunctionLinear Function

A linear function can be expressed in the form

( )f x mx b

where m and b are fixed numbers.

y mx b Equation notation

Function notation

13

Role of m and b in the Linear Function f (x) = mx + b.

The Role of m (slope)

f changes m units for each one-unit change in x.

The Role of b (y-intercept)

When x = 0, f (0) = b

14

The graph of a Linear Function: The graph of a Linear Function: Slope and Slope and yy-Intercept-Intercept

y-axis

x-axis

(1,2)

Ex. Sketch f (x) = 3x – 1

y-intercept

Slope = 3/1

15

Graphing a Line Using InterceptsGraphing a Line Using Intercepts

y-axis

x-axis

Ex. Sketch 2y + 3x = 6

y-intercept (x = 0)

x-intercept (y = 0)

16

SlopeSlope – – the slope of a non-vertical line the slope of a non-vertical line that passes through the points that passes through the points

is given by: is given by:

2 2,x y and 1 1,x y

2 1

2 1

y yym

x x x

Ex. Find the slope of the line that passes through the points (4,0) and (6, -3) 3 0 3 3

6 4 2 2

ym

x

17

Zero Slope; Undefined SlopeEx. Find the slope of the line that passes through the points (4,5) and (2, 5).

Ex. Find the slope of the line that passes through the points (4,1) and (4, 3).

5 5 00

2 4 2

ym

x

3 1 2

4 4 0

ym

x

UndefinedThis is a vertical line

This is a horizontal line

18

Point-Slope FormPoint-Slope Form

1 1

1 4 3

4 11

y y m x x

y x

y x

An equation of a line that passes through the point with slope m is given by:

1 1,x y

Ex. Find an equation of the line that passes through (3,1) and has slope m = 4

1 1y y m x x

19

**Two lines are parallel if and only if their slopes are equal or both undefined

Ex. Find an equation of the line that passes through (3,5) and is parallel to the line

21.

3y x

So m = 2/3 and we have the point (3, 5):

25 3

3y x

23

3y x

20

Horizontal LinesHorizontal Lines

y = 2

Can be expressed in the form y = b

21

Vertical LinesVertical Lines

x = 3

Can be expressed in the form x = a

22

Linear Models

Cost Function: ( )C x mx b

x = number of items

** m is the marginal cost (cost per item), b is fixed cost.

Revenue Function: ( )R x mx** m is the marginal revenue.

Profit Function: ( ) ( ) ( )P x R x C x

23

Break-Even AnalysisBreak-Even AnalysisThe break-even level of operation is the level of production that results in no profit and no loss.

Profit = Revenue – Cost = 0

Revenue = Cost

Dollars

Units

loss

Revenue

Cost

profit

Break-even point

24

Cost, Revenue, and Profit FunctionsCost, Revenue, and Profit FunctionsEx. A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find:

a. The cost function

b. The revenue function

c. The profit from 900 shirts

Cost: C(x) = 3x + 3600 where x is the number of shirts produced.

Revenue: R(x) = 12x where x is the number of shirts sold.

Profit: P(x) = Revenue – Cost

= 12x – (3x + 3600) = 9x – 3600

P(900) = 9(900) – 3600 = $4500

25

Cost: C(x) = 3x + 3600

Ex. A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find the break-even point.

If x is the number of shirts produced and sold

Revenue: R(x) = 12x( ) ( )

12 3 3600

400

R x C x

x x

x

(400) 4800R

At 400 units the break-even revenue is $4800

26

Linear DemandEx. The quantity demanded of a particular game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games. Find a demand equation relating the price p, and the quantity demanded, q (in units of 100).

( , ) : (6,50) and (10, 34)p q

34 50 164

10 6 4m

4 74q p 50 4( 6)q p

27

Market EquilibriumMarket EquilibriumMarket Equilibrium occurs when the quantity produced is equal to the quantity demanded.

q

p

supply curve

demand curve

Equilibrium Point

shortagesurplus

28

Ex. The maker of a plastic container has determined that the demand for its product is 400 units if the unit price is $3 and 900 units if the unit price is $2.50. The manufacturer will not supply any containers for less than $1 but for each $0.30 increase in unit price above the $1, the manufacturer will market an additional 200 units. Both the supply and demand functions are linear. Let p be the price in dollars, q be in units of 100 and find:

a. The demand function

b. The supply function

c. The equilibrium price and quantity

29

a. The demand function

, : 3, 4 and 2.5,9 ;p q9 4

102.5 3

m

4 10 3q p

10 34q p

b. The supply function

, : 1,0 and 1.3,2 ;p q2 0 20

1.3 1 3m

20 20

3 3q p

30

c. The equilibrium price and quantity

Solve 10 34q p 20 20

3 3q p and

simultaneously.

10 34 (1/ 3)(20 20)p p 30 102 20 20

2.44

p p

p

The equilibrium quantity is 960 units at a price of $2.44 per unit.

10(2.44) 34 9.6q

31

Linear Change over TimeA quantity q, as a linear function of time t:

( )q t mt b

Rate of change of q

Quantity at time t = 0

*If q represents the position of a moving object, then the rate of change is velocity.

32

Linear RegressionLinear RegressionThe method of least squares is to determine a straight line that best fits a set of data points when the points are scattered about a straight line.

least squares line

33

The Method of Least SquaresThe Method of Least Squares

22

n xy x ym

n x x

y m xb

n

Given the following n data points:

The least-squares (regression) line for the data is given by y = mx + b, where m and b satisfy:

and

1 1 2 2( , ), ( , ),..., ( , )n nx y x y x y

34

Ex. Find the equation of least-squares for the data (1, 2) (2,3) (3,7)

= 2.5

2.5 1y x

2

3 29 6 12

3 14 6m

12 2.5 6

3b

xx yy xyxy xx22

11 22 22 11

22 33 66 44

33 77 2121 99

Sum: 6 12 29 14

= –1

35

Coefficient of Correlation

A measurement of the closeness of fit of the least-squares line. Denoted r, it is between –1 and 1, the better the fit, the closer it is to 1 or –1.

2 22 2

n xy x yr

n x x n y y

36

Ex. Find the correlation coefficient for the least-squares line from the last example.

(1, 2) (2,3) (3,7)Points:

2 22 2

n xy x yr

n x x n y y

2 2

3 29 6 12

3 14 6 3 62 12

= 0.9449