Chapter 1Functions andLinear Models

Sections 1.3 and 1.4

Linear 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

Graph of a Linear Function

The graph of a linear function is a straight line.

( )y f x mx b

m is called the slope of the line and

b is the y-intercept of the line.

This means that we need only two points to completely determine its graph.

y-axis

x-axis

(1,2)

Example: Sketch the graph of f (x) = 3x – 1

y-intercept

Arbitrary point

(0,-1)

Role of m and b in 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

To see how f changes, consider a unit change in x.

( )f x mx b ( 1) ( 1)f x m x b

Then, the change in f is given by

( 1) ( )f x f x ( 1)m x b ( )mx b ( 1) ( )f x f x mx m b mx b ( 1) ( )f x f x m

Role of m and b in f (x) = mx + b

Role of m and b in f (x) = mx + b

Role of m and b in f (x) = mx + b

Role of m and b in f (x) = mx + b

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

y-axis

x-axis

(1,2)

Example: Sketch the graph of f (x) = 3x – 1

y-intercept

Slope = 3/1

Graphing a Line Using Intercepts

y-axis

x-axis

Example: Sketch 3x + 2y = 6

y-intercept (x = 0)

x-intercept (y = 0)

Delta Notation

2 1q q q

If a quantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Example: If x is changed from 2 to 5, we write

2 1 5 2 3x x x

Delta Notation

Example: the slope of a non-vertical line that passes through the points (x1 , y1) and (x2 , y2) is given by:

2 1

2 1

y yym

x x x

Example: 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

Delta Notation

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

Example: 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

ExamplesEstimate the slope of all line segments in the figure

Point-Slope Form of the Line

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 (x1 , y1) with slope m is given by:

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

1 1y y m x x

Horizontal Lines

y = 2

Can be expressed in the form y = b

Vertical Lines

x = 3

Can be expressed in the form x = a

Linear Models: Applications of linear

Functions

First,General Definitions

Cost Function

A cost function specifies the cost C as a function of the number of items x produced. Thus, C(x) is the cost of x items.

The cost functions is made up of two parts:

C(x)= “variable costs” + “fixed costs”

If the graph of a cost function is a straight line, then we have a Linear Cost Function.

If the graph is not a straight line, then we have a Nonlinear Cost Function.

Cost Function

Linear Cost Function

Dollars

Units

Cost

Dollars

Units

Cost

Dollars

Units

CostDollars

Units

Cost

Non-Linear Cost Function

Revenue Function

The revenue function specifies the total payment received R from selling x items. Thus, R(x) is the revenue from selling x items.

A revenue function may be Linear or Nonlinear depending on the expression that defines it.

Linear Revenue Function

Dollars

Units

Revenue

Nonlinear Revenue Functions

Dollars

Units

RevenueDollars

Units

Revenue

Profit Function The profit function specifies the net proceeds P.

P represents what remains of the revenue when costs are subtracted. Thus, P(x) is the profit from selling x items.

A profit function may be linear or nonlinear depending on the expression that defines it.

Profit = Revenue – Cost

Linear Profit Function

Dollars

Units

Profit

Nonlinear Profit Functions

Dollars

Units

ProfitDollars

Units

Profit

The Linear Models are

Cost Function: ( )C x mx b

** 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

where x = number of items (produced and sold)

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

To find the break-even point we set the profit function equal to zero and solve for x.

( ) 0

( ) ( ) 0

P x

R x C x

Break-Even Analysis

Profit = 0 means Revenue = Cost

Dollars

Units

loss

Revenue

Cost

profit

Break-even point

Break-even Revenue

The break-even point is the level of production that results in no profit and no loss.

Example: 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

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

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

P (x) = R(x) – C(x)

P (x) = 12x – (3x + 3600) = 9x – 3600

P(900) = 9(900) – 3600 = $4500

C (x) = R (x)

Example: 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.

The break even point is the solution of the equation

12 3 3600

400

x x

x

and (400) 4800R

Therefore, at 400 units the break-even revenue is $4800

-200 200 400 600 800 1000 1200 1400

-4000

-3000

-2000

-1000

1000

2000

3000

4000

5000

6000

7000

x = number of shirts sold

x

C, R, P in $

C(x)

-200 200 400 600 800 1000 1200 1400

-4000

-3000

-2000

-1000

1000

2000

3000

4000

5000

6000

7000

x = number of shirts sold

x

C, R, P in $

C(x)R(x)

-200 200 400 600 800 1000 1200 1400

-4000

-3000

-2000

-1000

1000

2000

3000

4000

5000

6000

7000

x = number of shirts sold

x

C, R, P in $

C(x)R(x)

P(x)

Demand Function

A demand function or demand equation expresses the number q of items demanded as a function of the unit price p (the price per item).

Thus, q(p) is the number of items demanded when the price of each item is p.

As in the previous cases we have linear and nonlinear demand functions.

Linear Demand Function

q = items demanded

Price p

Nonlinear Demand Functions

q = items demanded

q = items demanded

Price p Price p

Supply Function

A supply function or supply equation expresses the number q of items, a supplier is willing to make available, as a function of the unit price p (the price per item).

Thus, q(p) is the number of items supplied when the price of each item is p.

As in the previous cases we have linear and nonlinear supply functions.

Linear Supply Function

q = items supplied

Price p

Nonlinear Supply Functions

q = items supplied

q = items supplied

Price p Price p

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

q

p

supply curve

demand curve

Equilibrium Point

shortage surplus

q

p

shortage

supply curve

demand curve

surplus

Equilibrium price

Equilibrium demand

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

Market Equilibrium

To find the Equilibrium price set the demand equation equal to the supply equation and solve for the price p.

To find the Equilibrium demand evaluate the demand (or supply) function at the equilibrium price found in the previous step.

Market Equilibrium

Example of Linear DemandThe quantity demanded of a particular computer game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games.

1 1 2 2( , ) (6,50) and ( , ) (10, 34)p q p q

2 1

2 1

34 50 164

10 6 4

q qqm

p p p

4 74q p 50 4( 6)q p

Find a linear demand equation relating the price p, and the quantity demanded, q (in units of 100).

Example: 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. Assume that 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 equilibrium demand

a. The demand function 1 1 2 2, 3, 4 and , 2.5,9 ;p q p q

9 410

2.5 3

qm

p

4 10 3q p

10 34q p

b. The supply function

1 1 2 2, 1,0 and , 1.3,2 ;p q p q

2 0 20

1.3 1 3

qm

p

20 20

3 3q p

c. The equilibrium price and equilibrium demand

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

2.44

p p

p

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

10(2.44) 34 9.6q

Demand Supply

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

( )q t mt b

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

Rate of change of q

Quantity at time t = 0

Linear Regression

We have seen how to find a linear model given two data points. We find the equation of the line passing through them.

However, we usually have more than two data points, and they will rarely all lie on a single straight line, but may often come close to doing so.

The problem is to find the line coming closest to passing through all of the points.

Linear Regression

We use the method of least squares 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

The 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

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

The scatter plot of the points is

Solution: We complete the following table

2.5 1y x

2

3 29 6 122.5

3 14 6m

12 2.5 61

3b

x y xy x2

1 2 2 1

2 3 6 4

3 7 21 9

Sum: 6 12 29 14

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

The scatter plot of the points and the least squares line is

2.5 1y x

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

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

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

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