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Linear and Quadratic FunctionsOn completion of this module you should be able to: define the terms function, domain, range, gradient,

independent/dependent variable use function notation recognise the relationship between functions and

equations graph linear and quadratic functions calculate the function given initial values (gradient, 1 or 2

coordinates) solve problems using functions model elementary supply and demand curves using

functions and solve associated problems

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A function describes the relationship that exists between two sets of numbers.

Put another way, a function is a rule applied to one set of numbers to produce a second set of numbers.

Functions

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Example: Converting Fahrenheit to Celsius

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C F

This rule operates on values of F to produce values of C.

The values of F are called input values and the set of possible input values is called the domain.

The values of C are called output values and the set of output values produced by the domain is called the range.

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Consider the function3

)(

x

xxf

The x are the input values and f(x), read f of x, are the output values.

The domain is the set of positive real numbers including 0 and excepting 3. (Why?) The output values produced by the domain is the range.

Sometimes the symbol y is used instead of f(x).

Function Notation

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An equation is produced when a function takes on a specific output value.

eg f(x) = 3x + 6 is a function.

When f(x) = 0, then the equation becomes

0 = 3x + 6

which can be easily solved (to give x = −2)

Function and Equations

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0)( xf

)(xf

x

)6,0(

)0,2(

This is shown graphically as follows:

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Input and output values form coordinate pairs: (x, f(x)) or (x, y).

x values measure the distance from the origin in the horizontal direction and f(x) values the distance from the origin in the vertical direction.

To plot a straight line (linear function), 2 sets of coordinates (3 sets is better) must be calculated. For other functions, a selection of x values should be made and coordinates calculated.

Graphing Functions

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Graph f(x) = 2x − 4

)( xfx

)10,3( 104)3(2 3 )4,0( 44)0(2 0

2) (3, 24)3(2 3

Example: Linear Function

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3 3

4

2

42)( xxf

x

)(xf

10

2( ) 2 5 2f x x x

At the y-intercept, x = 0, so

and the coordinate is (0,2).

2( ) 2 0 5 0 2 2f x

Example: Quadratic Function

Graph the function:

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At the x-intercept, f(x) = 0, so

and the coordinates are (2,0) and (0.5,0).

2520 2 xx

25 ( 5) 4 2 22 2

x

435

0.5or 2

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2bxa

51.25

2(2)

When 1.25,x 2( ) 2(1.25) 5(1.25) 2 1.125f x

Vertex:

The coordinates of the vertex are: (1.25, −1.125).

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1

2

2

-1

)(xf252)( 2 xxxf

x

(1.25, -1.125)

(2,0)

(0.5,0)

(0,2)

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All linear functions (or equations) have the following features:

a slope or gradient (m) a y-intercept (b)

If (x1, y1) and (x2, y2) are two points on the line then the gradient is given by:

Linear Functions

2 1

2 1

y ymx x

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Gradient is a measure of the steepness of the line. If m > 0, then the line rises from left to right. If m < 0, the line falls from left to right. A horizontal line has a gradient of 0; a vertical line has an undefined gradient. The y-intercept is calculated by substituting x = 0 into the equation for the line.

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All straight line functions can be expressed in the form y = mx + b

Note: The standard form equation for linear functions is Ax + By + C = 0. Equations in this form are not as useful as when expressed as y = mx + b.

Equations can be derived in the following way, depending on what information is given.

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1. Given (x1, y1) and (x2, y2)

2. Given m and (x1, y1)

3. Given m and b

12

12

1

1

xxyy

xxyy

Deriving Straight Line Functions

1 1( )y y m x x

y mx b

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A tractor costs $60,000 to purchase and has a useful life of 10 years.

It then has a scrap value of $15,000.

Find the equation for the book value of the tractor and its value after 6 years.

Problem: Depreciation

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V

?

6

15,000

10 t

60,000

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Value (V) depends on time (t).

t is called the independent variable andV the dependent variable.

The independent variable is always plotted on the horizontal axis and the dependent variable on the vertical axis.

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1 1

When 0, 60,000 (0, 60,000) ( , )

t Vx y

2 2

When 10, 15,000 (10, 15,000) ( , )

t Vx y

60,000 15,000 60,0000 10 0

yx

Given two points, the equation becomes:

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60,000 4500yx

60,000 4500y x

When 6, 4500 6 60,000 33,000t V

The book value of the tractor after 6 years is $33,000.

4500 60,000V t

4500 60,000y x

or more correctly

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Suppose a manufacturer of shoes will place on the market 50 (thousand pairs) when the price is $35 (per pair) and 35 (thousand pairs) when the price is $30 (per pair).

Find the supply equation, assuming that price p and quantity q are linearly related.

) ,()30 ,35(

) ,()35 ,50(

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1 1

yx

yx

Example

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11

2 2

(50, 35) ( , )

(35, 30) ( , )

x y

x y

50353530

5035

xy

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5035

xy

1 2 1

1 2 1

y y y yx x x x

)50(3135 xy

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35 0.33 16.67y x

0.33 16.67 35y x

0.33 18.33y x

0.33 18.33p q

The supply equation is

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For sheep maintained at high environmental temperatures, respiratory rate r (per minute) increases as wool length l (in centimetres) decreases.

Suppose sheep with a wool length of 2cm have an (average) respiratory rate of 160, and those with a wool length of 4cm have a respiratory rate of 125.

Assume that r and l are linearly related.

(a) Find an equation that gives r in terms of l.

(b) Find the respiratory rate of sheep with a wool length of 1cm.

Example

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1 1

2 2

(2,160) ( , )(4,125) ( , )

x yx y

5.1724160125

12

12

xxyy

m

(a) Find r in terms of l

l is independent

r is dependent

Coordinates will be of the form: (l, r).

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1 1( )y y m x x

160355.17 xy1955.17 xy

17.5 195r l

(2, 160), (4, 125)

160 17.5( 2)y x

17.5m

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17.5(1) 195177.5

r

When wool length is 1 cm, average respiratory rate will be 177.5 per minute.

(b) When l = 1

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All quadratic functions can be written in the form

where a, b and c are constants and a 0.

2( )f x ax bx c

Quadratic Functions

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In general, the higher the price, the smaller the demand is for some item and as the price falls demand will increase.

Demand curve

q

p

Elementary Supply and Demand

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Concerning supply, the higher the price, the larger the quantity of some item producers are willing to supply and as the price falls, supply decreases.

q

p

Supply curve

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Note that these descriptions of supply and demand imply that they are dependent on price (that is, price is the independent variable) but it is a business standard to plot supply and demand on the horizontal axis and price on the vertical axis.

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The supply of radios is given as a function of price by

and demand by

Find the equilibrium price.

22 8 12, 2 5S p p p p

2 18 68, 0 5D p p p p

Example: Equilibrium price

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Graphically,

Note the restricted domains.

1 2 3 4 5 p

70equilibrium price

00

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Algebraically, D(p) = S(p)

12826818 22 pppp

012688182 22 pppp056102 pp

56 ,10 ,1 cba

210 ( 10) 4( 1)(56)2 1

p

4or 14p

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−14 is not in the domain of the functions and so is rejected.

The equilibrium price is $4.

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If an apple grower harvests the crop now, she will pick on average 50 kg per tree and will receive $0.89 per kg.

For each week she waits, the yield per tree increases by 5 kg while the price decreases by $0.03 per kg.

How many weeks should she wait to maximise sales revenue?

Example: Maximising profit

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Weight and Price can both be expressed as functions of time (t).W(t) = 50 + 5tP(t) = 0.89 − 0.03t

)()(Price Weight Revenue

tPtW

)03.089.0)(550( tt 215.045.45.15.44 ttt

5.4495.215.0 2 tt

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Maximum occurs at

83.9)15.0(2

95.22

abt

She should wait 9.83 weeks ( 10 weeks) for maximum revenue.(R = $59 per tree)

0.15 0a