1 Linear and Quadratic Functions On 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
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1 Linear and Quadratic Functions On completion of this module you should be able to: define the terms function, domain, range, gradient, independent/dependent.
3 Example: Converting Fahrenheit to Celsius 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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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
5 329
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
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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
25
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)