1
1
Equations
and functions
ALGEBRAIC MODELLING
Jenni is comparing two Internet access plans. Optnet has a monthly access fee of $8 and charges 60 cents per hour of Internet use. OzExpress charges $1 per hour of Internet use but has no monthly access fee. The two plans are
represented by the formulas Optnet: C = 0.6t+ 8 OzExpress: C =t
where C is the cost in dollars and t is the number of hours of Internet use.
Can you advise Jenni? What will you tell her?
This problem involves analysing and comparing functions and formulas, which are the main themes of this chapter. Such a problem may be solved:
algebraically, by solving equations
graphically, on graph paper or using a graphics calculator or graphing software, or by a ‘guess, check and refine’ method, using a table of values, spreadsheet or calculator.
In this chapter you will learn how to:
add, subtract, multiply and divide algebraic terms calculate with numbers in scientific notation solve different types of linear equations
solve equations involving powers, including the use of ‘guess, check and refine’ substitute values into formulas and solve equations for a variety of practical problems change the subject of a formula
examine real life situations that can be modelled by linear functions draw a line of best fit to a set of empirical data and find its equation
interpret the point of intersection of the graphs of two linear functions drawn from practical contexts, including ‘break-even’ points.
ALGEBRAIC EXPRESSIONS
An algebraic expression is a general statement involving pronumerals or variables. It is made up of terms. For example, 2p2+ 7p− 4 has three terms: 2p2, 7p and −4.
Types of algebraic expressions
When writing algebraic expressions, terms with higher powers are usually written first. For example, we write 8t3− 5t2− 1, not −5t2− 1 + 8t3. Because of this, the term with the highest
power is called the leading term. The leading term of 8t3− 5t2− 1 is 8t3.
An algebraic expression is classified by the power of its leading term:
8t3− 5t2− 1 is called a cubic expression because its leading term has a power of 3 (the variable is ‘cubed’).
4y2 −y+ 5 is called a quadratic expression because its leading term has a power of 2
(the variable is ‘squared’). Quad means ‘square’ in Latin.
2n− 1 is called a linear expression because its leading term has a power of 1. It is ‘linear’ because, when graphed as a function on the number plane, the graph is a straight line.
Simplifying algebraic expressions
Example 1
Simplify these expressions.
(a) 3d+ 8d2− 3 −d (b) 4xz+ 4x2+x2− 2z (c) −5k× 4k2t (d) 18mp2 45m2p
Solution
(a) 3d+ 8d2− 3 −d=3d−d+ 8d2− 3 Collecting like terms 3d and −d
= 2d+ 8d2− 3
= 8d2+ 2d− 3
(b) 4xz+ 4x2+x2− 2z=4x2+x2+ 4xz− 2z Collecting like terms 4x2 and x2 = 5x2+ 4xz− 2z
(c) −5k× 4k2t =−5 × 4 ×k×k2 ×t
=−20k3t
(d) 18mp2 45m2p=
=
Expanding algebraic expressions
Example 2
Expand and simplify these expressions.
(a) 3(2m+ 7) − 2(m2+ 4) (b) r(4r− 10) − (4r− 10)
Solution
(a) 3(2m+ 7) − 2(m2+ 4)= 6m+ 21 − 2m2− 8
=−2m2+ 6m+ 13
(b) r(4r− 10) − (4r− 10)= 4r2− 10r− 4r+ 10
= 4r2 − 14r+ 10
18m p2
45m2p
---2p 5m
1. Write these algebraic expressions in the correct order, then classify them as being linear (L), quadratic (Q) or cubic (C).
(a) 3 − 4x+ 6x3 (b) 10 − 2x (c) 5 +x2−x
(d) 8 + x (e) 7x−x3+x2 (f) 1 + 20x
(g) 7 + 5x2 (h) 2 − x (i) x− x2+ 1
(j) x+ 7x2−x3 (k) −5x2+x3+ 14 (l) 4 −x 2. Simplify these expressions.
(a) x3+ 2x2+ 5x2−x3 (b) 4m+ 7 + 18m− 5 (c) 3bd+ 4d+d2− 5d (d) 6c− 6 −c+ 20
(e) 2r+r+ 2r2−r (f) 7tu+t2− 8t2+tu
(g) −3e+ 9 +e+e2 (h) 5p+ 4p− 9p2+ 7p2
(i) 2xy+ 3yz+ 4xy− 8yz (j) 7k+ 11y− 20 + 3k− 6 (k) 16j− 3jk− 5j− 4jk (l) 8u2+ 6 − 10 − 5u2
3. Simplify these expressions.
(a) 3m2× 2mn (b) −5p×−6p (c) (−2rt2)2
(d) 16w5y2 8wy (e) 10u2× (f)
(g) 4p2 10p (h) 6r×−3r2 (i) (3x2)3
(j) 5u×u (k) 5uu (l)
4. Expand and simplify these expressions.
(a) 3(k+ 4) + 4(k+ 4) (b) 2(d+ 1) − 2(d− 1) (c) 5(r+ 3) + 4(r+ 10) (d) 8(m+ 5) − 3(m− 5) (e) 2(5a+ 6) + (a− 1) (f) 3(2t+ 1) − (4t+ 8) (g) x(x+ 7) − 5(x− 3) (h) 5u(2 −u) +u(6u+ 3) (i) 2πr(r+h) −πr2 (j) 2x2(7 −x) +x(x− 1)
(k) 4(4m+ 1) − 3(3m− 2) (l) 8(5 − 2r) +r(3r+ 1) (m) 6(2d− 3) + (2d− 3) (n) f(5f+ 5) + 5(f+ 5) (o) 10x(x− 1) +x(5x+ 4) (p) 2c(2c+ 7) + 2c(4 −c) 5. Simplify these expressions.
(a) −3b2× (b) (−4ab)2 (c) 5x+ 4xz− 4x+ 6
(d) 3w− 12y+ 2y2− 4w (e) 8kp2 12k (f) 4de× 5d2e
(g) (h) 6jr+ 2r− 4jr− 5r (i) 6wz×
(j) 8tu+ 4u2−u2− 8t (k) 18tu 2tu2 (l) (5f3)2
(m) 8a2y× y2 (n) 4ay+y−ay+y (o) 2mn+n−n+m2
(p) −6hk×−2k2 (q) (r)
Exercise 1-01:
Algebraic expressions
1 2
---3 4
--- 1
3
---3u 2
--- 18k3y
3k y2
---6d e2
14d2e2
---4bd 6
---10ut –
4t2
--- 5w2
2z
---1 2
---15d y2
20y2
--- –24a2b2 16a – b2
---SCIENTIFIC NOTATION
Scientific notation (also called standard form) is a shorthand for writing very large numbers or very small numbers using powers of 10.
Large numbers are written with positive powers of 10. Small numbers are written with negative powers of 10.
Example 3
Express these numbers in scientific notation.
(a) 13 700 000 000 (b) 20 480 (c) 0.000 000 61
Solution
(a) 13 700 000 000 = 1.37 × 1010 10 places after the first significant figure, 1
(b) 20 480= 2.048 × 104 4 places after the first significant figure, 2
(c) 0.000 000 61 = 6.1 × 10−7 7 places up to and including the first significant figure, 6
Example 4
Express these numbers in normal decimal form.
(a) 8.5 × 104 (b) 8.5 × 10−4 (c) 9.31 × 10−6
Solution
(a) 8.5 × 104= 85 000 4 places after the first significant figure, 8
(b) 8.5 × 10−4= 0.000 85 4 places up to and including the 8
(c) 9.31 × 10−6= 0.000 00931 6 places up to and including the 9
Example 5
Evaluate the following.
(a) (7.4 × 105) − (8.3 × 104) (b) (c)
Solution
Calculator keys
(a) (7.4 × 105) − (8.3 × 104) = 657 000 7.4 5 8.3 4
(b) = 9 × 109 or 9 000 000 000 2.7 6 3 4
(c) = 650 000 4.225 11
Scientific notation has the form
m× 10n
where m is a number between 1 and 10 and n is an integer.
An integer is a positive or negative whole number, or zero.
2.7×106
3×1 0–4
--- 4.225×1011
EXP − EXP =
2.7×106
3×1 0–4
--- EXP ÷ EXP +/− =
4.225×1011
1. Express these numbers in scientific notation.
(a) 230 000 (b) 8 950 000 (c) 0.000 17
(d) 0.006 (e) 20 000 000 (f) 0.0308
(g) 17 300 (h) 902 (i) 0.24
(j) 0.000 036 (k) 4 800 000 (l) 700 000 000
2. Express these numbers in normal decimal form.
(a) 7.2 × 105 (b) 7.2 × 10−3 (c) 6 × 104
(d) 8.19 × 107 (e) 2 × 10−6 (f) 3.2 × 10−5
(g) 9.6 × 108 (h) 4 × 106 (i) 1.75 × 10−2
(j) 2.8 × 103 (k) 3.087 × 10−4 (l) 5.129 × 107
3. Evaluate:
(a) (3.6 × 105) + (1.1 × 103) (b) (2.7 × 10−4)2
(c) (5.46 × 107) + (8.2 × 104) (d)
(e) (f)
(g) (7.7 × 104)3 (h) (6.2 × 10−4) − (4.11 × 10−5) 4. Evaluate, giving your answer in scientific notation correct to 2 significant figures:
(a) (8.4 × 105)2 (b) (3.905 × 107) + (1.1 × 105)
(c) (d)
(e) (8.6 × 108) × (9.4 × 10−3) (f) (3.98 × 103)3
(g) (h)
FORMULAS
Example 6
Heron’s formula for calculating the area of a triangle with side lengths a, b, c is
A= where s= (a+b+c)
Use the formula to find (correct to 1 decimal place) the area of a triangle with sides of length 5 cm, 8 cm and 10 cm.
Solution
Let a= 5, b= 8, c= 10.
s= (5 + 8 + 10) = 11.5
A= =
= = 19.8100 …
Area ≈ 19.8 cm2
Exercise 1-02:
Scientific notation
4.5×10–4
6×1 0–6
---1.089×10–9 1
8×1 06
---5.8×108 2.7×108
4.11× 05
---1 5.4×10–9
--- 3 6×1 012
a
b
c s s( –a)(s–b)(s–c) 1
2
---1 2
---11.5 ---11.5( –5)(11.5–8)(11.5–10) 11.5×6.5×3.5×1.5
Example 7
The time (in seconds) it takes a swing to go back and forth once is
T= 2π
where l is the length of the swing (in metres) and g is the gravitational acceleration. Find T correct to 1 decimal place if l= 2.8 m and g= 9.8 m/s2.
Solution
T= 2π
= 3.3585 … ≈ 3.4 s
1. The formula for the surface area of a closed cylinder is S= 2πr(r+h)
where r is the radius of its base and h is its height. Calculate (correct to 2 significant figures) the surface area of a cylinder with radius 0.8 m and height 2.3 m.
2. The kinetic energy K (in joules, J) of an object of mass m kg travelling at speed v m/s is
K= mv2
Calculate the kinetic energy of an object of mass 2.5 kg travelling at 4 m/s.
3. The value of an item depreciating over time is S=V(1 −r)n
where V is its original value, r is the annual rate of depreciation expressed as a fraction or decimal, and n is the number of years. Calculate the value of a $4200 computer after 3 years if it is depreciating at 27% p.a. Answer to the nearest dollar.
4. The body mass index (BMI) of an adult is
B=
where m is the mass in kilograms and h is the height in metres.
(a) Glen is 1.83 m tall and weighs 88 kg. Calculate his body mass index correct to 1 decimal place.
(b) If a BMI between 20 and 25 is an indication of good health, can Glen improve his health? If so, how?
5. The formula for converting Australian dollars ($A) to New Zealand dollars ($NZ) is NZ = 1.2831A. Convert the following $A amounts to $NZ, correct to the nearest cent.
(a) $A40 (b) $A84.95 (c) $A12.20
6. The number of days fresh milk will keep if stored at temperature T°C is
D=
How many days will fresh milk keep if it is stored at:
(a) 3°C? (b) 5°C? (c) freezing point (0°C)?
7. The size of each angle (in degrees) in a regular polygon with n sides is
a= 180 − Calculate the size of each angle in:
(a) a regular nonagon (9 sides) (b) a regular decagon (10 sides) l
g
---2.8 9.8
---Exercise 1-03:
Formulas
1 2
---m h2
---6 T +1
---8. Given that E=mc2, calculate the value of E when m= 0.002 and c= 3 × 108.
9. The number of times per minute a cricket chirps on a hot summer’s night is n= 8T− 24
where T is the air temperature in °C.
(a) How many times per minute will a cricket chirp if the temperature is 22°C? (b) This formula is an example of an algebraic model. According to this model, will the
frequency of chirping increase or decrease as the night gets hotter?
10. The area of a trapezium is given by the formula
A= (a+b)h
where a and b are the lengths of the parallel sides and h is the distance between them. Calculate the area of the trapezium illustrated.
11. The velocity V m/s required for a rocket to escape the Earth’s gravity is V=
where g= 9.8 m/s2 (the gravitational acceleration) and r= 6.38 × 106 m (the radius of
the Earth). Calculate the escape velocity of the rocket correct to 3 significant figures.
12. When prescribing medicine for infants, the dosage is given by the formula
D= (Fried’s rule) or D= (Young’s rule)
where A is the adult dosage, m is the infant’s age in months and y is the infant’s age in years. Jessie, aged 24 months, needs to take cough medicine with a recommended adult dosage of 12 mL. Calculate Jessie’s dosage, correct to 2 decimal places, using:
(a) Fried’s rule (b) Young’s rule
13. The blood alcohol content (BAC) of a person who consumes alcoholic drinks is BAC = 0.0012na
where n is the number of drinks taken and a is the amount of alcohol in each drink measured in millilitres. Sally drank three stubbies (250 mL each) of light beer (2.7% alcohol). Calculate:
(a) the amount of alcohol in each drink (b) Sally’s BAC level
14. The air temperature, T °C, outside an aeroplane is given by the formula T= 15 − 0.006h
where h is the height of the plane above sea level. Calculate correct to 1 decimal place the temperature outside the plane at height:
(a) 800 m (b) 975 m
15. The area of this irregular figure can be approximated using the formula
A≈ (a+ 4b+c) (Simpson’s rule) Calculate the approximate area of the figure if h= 18 m, a= 21 m, b= 25 m and c= 28 m.
6 cm
10 cm 7 cm
1 2
---2gr
mA 150
--- yA
y+12
---h h
a b c
---16. A country’s population density (in persons per square kilometre) is given by the formula
D=
where P is the population and A is the area in square kilometres. Calculate the population density of Australia, correct to 3 significant figures, if its population is 1.92 × 107 and
its area is 7.69 × 106 km2.
17. The area of this triangle is given by the formula
A= ab sin
Find A to 2 decimal places if a= 7 cm, b= 8 cm and θ= 51°.
18. The distance (in kilometres) that an observer can see to the horizon from the top of a structure of height h m is
d= 8
What distance can be seen from the top of the Eiffel Tower in Paris, a height of 320 m?
19. The volume of a sphere is given by the formula
V= πr3
Calculate the volume of the illustrated sphere correct to 2 significant figures.
20. Elena earns a taxable income of $57 821. Calculate her income tax using the formula T= 11 380 + 0.42(I− 50 000)
where T is the tax payable and I is the taxable income.
21. The area of an annulus (doughnut shape) is A=π(R2−r2)
where R is the longer radius and r is the shorter radius. Calculate the area of the illustrated annulus correct to 2 decimal places.
22. The price of an item n years ago before inflation was
P=
where A is its current price and r was the annual rate of inflation over this period written as a decimal. Use the formula to calculate (to the nearest $100) the price of a $27 000 car 8 years ago if the inflation rate during this time was 1.2%.
23. The volume (in cubic centimetres) of wood in a tree is V= 0.4724d2h+ 9.86
where d is the diameter of the trunk and h is the vertical distance between the trunk and the lowest branch. All dimensions are in centimetres.
(a) Calculate the volume V if d= 55 cm and h= 210 cm.
(b) Express this volume in cubic metres correct to 1 decimal place. P
A
---θ
b a
1 2
---h 5
---4.1 cm 4
3
---8 cm
3 cm
A 1+r
( )n
---SOLVING EQUATIONS
Example 8
Solve these equations.
(a) = (b) − = 4
Solution
(a) =
5(8x) = 7(2x− 1)
40x= 14x− 7 Expanding
26x=−7 Subtracting 14x from both sides
x= − Dividing both sides by 26
Just for the record
B
ODY MASS INDEXThe body mass index, calculated by the formula
B =
where m is mass in kg and h is height in m, was adopted by the World Health Organisation (WHO) in 1997 as an international standard of health and fitness. It is a convenient measure that represents the health of an individual by a single value, applicable to both male and female adults, and does not require reference to height–weight charts. The table interprets a range of BMI scores.
According to this measure, about 40% of Australians and 59% of Americans are classified as being overweight or obese. People who fall into this category have a greater risk of heart disease, stroke, diabetes, higher blood pressure and higher cholesterol.
Note, however, that the BMI does not take into account individual differences in frame size, muscle mass or distribution of body fat. The BMI is an example of an algebraic model and, like all mathematical models, it has some limitations. For example, the BMI cannot be used to measure the fitness levels of the following types of people:
body builders pregnant women
growing children below the age of 19 the frail and sedentary elderly.
Give a reason why the BMI does not apply to each of these types. m
h2
--- Health status BMI
Severe starvation Mild starvation Good health Ideal health Overweight Obesity Severe obesity
16–17 18–19 20–25 21–23 26–29 30–39 40 and above
8x 7
--- 2x–1 5
--- 4t
3
--- t+2 10
---8x 7
--- 2x–1 5
---Cross-multiplying:
(equivalent to multipying both sides by 35) 8x
7
--- 2x–1 5
---7 26
(b) − = 4
− = 30(4) Multiplying each term by 30
10(4t) − 3(t+ 2)= 120
40t− 3t− 6= 120 Expanding
37t− 6= 120 Collecting like terms
37t= 126 Adding 6 to both sides
t= (or 3 ) Dividing both sides by 37
Solve these equations.
1. 3(p− 4) =−27 2. 4x+ 4 =x− 11
3. 12d= 8 + 8d 4. = 29
5. = 6. = 4
7. −4(2k+ 1) = 0 8. 3c+ 8 = 2(c− 10)
9. 2 + =−8 10. 6 − 2k= 10
11. =−4 12. 6(2 − 3w) =−24
13. = 14. =
15. + 7 = − 4 16. 10k− 11 = 4(2k− 10)
17. 8c+ 16 =c− 12 18. =
19. = 3 20. 8(4 − 2c) =−16
21. =−2 22. =
23. =−5 24. 10y− 8 = 46 − 2y
25. = 26. 9r+ 24 = 3r
27. 5(d+ 5) = 2(2d+ 18) 28. =−3
29. 12g− 11 = 4(5g+ 7) 30. − 1 = + 6 4t
3
--- t+2 10
---30
10 4t
3 --- 1 30
3 t+2
10 --- 1 126 37 --- 15 37
---Exercise 1-04:
Solving equations
5h–2 2 ---2c 3 --- 2 9
--- 3r
7 --- r 5 ---– t 3
---6b–7 2 ---h 3 --- h 4 ---+ 3 4
--- 3y
5
--- y+1 2
---a 5
--- 2a 3
---4q–3 4
--- q+8 10
---m 3 ---- 3m
10 ---–
3d–1 5
--- 4d+7 10
---+ 3
5
--- x+1 12
---EQUATIONS INVOLVING POWERS AND ROOTS
Squares and cubes, square and cube roots
Example 9
Solve these equations.
(a) = 2 (b) w3+ 6 = 5 (c) = 7 (d) = 4
Solution
(a) = 2
x2− 13 = 36 Multiplying both sides by 18
x2= 49 Adding 13 to both sides
x=± Taking the square root of both sides
=±7
Note: Equations involving a pronumeral ‘squared’ (x2) usually have two answers.
(b) w3+ 6= 5
w3=−1 Subtracting 6 from both sides
w= Taking the cube root of both sides
=−1
Note: Equations involving a pronumeral ‘cubed’ (w3) only have one answer.
(c) = 7
= 72 Squaring both sides
4d− 9 = 49
4d= 58 Adding 9 to both sides
d= Dividing both sides by 4
= (or 14 )
(d) = 4
= 43 Cubing both sides
3p= 64
p= (or 21) Dividing both sides by 3
Power equations and the ‘guess, check and refine’ method
Example 10
In the following equations, the unknown pronumeral is a power. Solve them using the ‘guess, check and refine’ method, giving answers correct to 1 decimal place where appropriate. (a) 4x= 4096 (b) (2.2)k= 5 (c) 3r= 1.5
x2–13
18
--- 4d–9 3 3p
x2–13
18
---49
1 –
3
4d–9
4d–9
( )2
58 4
---29 2
--- 1 2
---3p
3
3p
3
( )3
64 3
---Solution
The ‘guess, check and refine’ method involves guessing the answer, checking it, then refining (improving) the guess until the answer (or a good approximation) is found. (a) 4x = 4096
The solution to 4x= 4096 is x= 6.
(b) (2.2)k= 5
As 2.22= 4.84 and 2.22.05≈ 5.0346, k must be between 2 and 2.05, so k≈ 2.0
(correct to 1 decimal place).
Note: The solution to this equation is only an approximation. (c) 3r = 1.5
r must be between 0.35 and 0.4, so r≈ 0.4 (correct to 1 decimal place).
Equations of the form ax=b can also be solved using , the logarithmic function key
on your calculator, using the formula x=
For example, to solve the equation 1.2x= 5, evaluate x= by pressing the following
keys on a calculator:
Display
5 1.2
(or 5 1.2 on older calculators)
The solution is x≈ 8.8 (correct to 1 decimal place). Checking: 1.28.8≈ 4.975 02 ≈ 5.
Guess Check Result
x= 8 48= 65 536 Too high
x= 4 44= 256 Too low
x= 6 46= 4096 Correct
Guess Check Result
k= 2 2.22= 4.84 Too low
k= 2.5 2.22.5= 7.1788 … Too high
k= 2.2 2.22.2= 5.6666 … Too high
k= 2.12.22.1= 5.2370 … Too high
k= 2.05 2.22.05= 5.0346 … Too high
Guess Check Result
r= 131= 3 Too high
r= 0.5 30.5= 1.7320 … Too high
r= 0.4 30.4= 1.5518 … Too high
r= 0.3 30.3= 1.3903 … Too low
r= 0.35 30.35= 1.4689 … Too low
Calculator keys
4 xy 8 =
Technology:
The
logkey on your calculator
log
logb loga
---log 5 log 1.2
---log log = 8.8274 …
Another way of finding an unknown power (or a close guess) in a power equation is to use the repeated (or constant) multiplication feature of your calculator. For example, to solve 4x= 4096, we need to multiply 4 by itself repeatedly until the display shows 4096.
To do this, first press: Display
4 4 (‘count 1’ for 41)
or 4 on older calculators.
Then press repeatedly to continue multiplying by 4:
(‘count 2’ for 42) (‘count 3’ for 43)
(‘count 4’ for 44)
Count the number of times you press the key, including the very first time. In this example, has been pressed 4 times, so 44= 256. Press a further 2 times and you
will discover that 46= 4096. So the solution to the equation 4x= 4096 is x= 6.
1. Solve these equations.
(a) = 7 (b) 3m2+ 5 = 80 (c) 5u3=−40
(d) = 3 (e) −3 + = 9 (f) = 18
(g) =−6 (h) 4k3+ 11 = 267 (i) =
2. Solve these equations, expressing answers correct to 2 decimal places. (a) d2= 96 (b) 5y2− 10 = 14 (c) 3c3= 20
(d) = 7 (e) p3+ 12 = 6 (f) −7w2=−10
(g) 8h3− 12 = 18 (h) = 9 (i) p3= 6
3. Solve these equations, correct to 1 decimal place where appropriate.
(a) 3r = 531441 (b) 6x= 279 936
(c) 10p= 10 000 (d) 2d= 32 768
(e) 5y= 1 953 125 (f) 1.5a= 7.593 75
(g) 2t = 2000 (h) 5c= 63 470
(i) 3w= 945 (j) 1.7n= 5
(k) 2.5h= 88 (l) 1.06h= 4
4. Use the key on your calculator and the formula x= to solve these equations, correct to 1 decimal place where appropriate.
(a) 2h= 65 536 (b) 5k= 48 828 125 (c) 4u= 16 384
(d) 7n= 16 807 (e) 3p= 128 (f) 2r= 89
(g) 1.07x= 2 (h) 1.5t= 3 (i) 1.1a= 2.2
Technology:
Using the repeated multiplication feature of your calculator
= ANS × 4
× ×
=
= 16
= 64
= 256
=
= =
Exercise 1-05:
Equations involving powers and roots
5y+4
2a+1
3 y d2
8
---2e
3 x
5 --- 4
10
---n2
4
---r2+4
5
--- 1
4
---log logb
---The spreadsheet (or List feature of a graphics calculator) is a useful tool for implementing the ‘guess, check and refine’ method of solving equations. Create a spreadsheet that allows you to solve equations of the form ax=b by calculating values of ax for different values of x.
The value of a can be entered, followed by a starting guess for x and a step size for increasing the values of x. See the example below for solving the equation 1.2x= 5.
1st guess: values of x from 2 to 12 2nd guess: values of x from 8 to 9
From the first-guess spreadsheet, x lies between 8 and 9 (closer to 9). From the second-guess spreadsheet, x lies between 8.8 and 8.9 (closer to 8.8). So the value of x in 1.2x= 5 is 8.8
(correct to 1 decimal place).
CHANGING THE SUBJECT OF A FORMULA
If a moving object has (initial) speed u m/s and acceleration a m/s2, its final speed
v m/s after time t seconds is given by the formula v=u+at
This formula has four variables: v, u, a and t. Because the formula is written for v, with v on the left hand side of the ‘=’ sign, we say that v is the subject of the formula. However, a formula can be rearranged so that one of its other variables becomes the subject.
Example 11
For the formula v=u+at, make:
(a) u the subject (b) t the subject
A B
1 a 1.2
2 Starting guess, x 2
3 Step size 1
4
5 x ax
6 2 1.44
7 3 1.728
8 4 2.0736
9 5 2.48832
10 6 2.985984
11 7 3.5831808
12 8 4.29981696
13 9 5.159780352
14 10 6.1917364224
15 11 7.43008370688
16 12 8.916100448256
Spreadsheet activity:
Guess, check and refine
A B
1 a 1.2
2 Starting guess, x 8
3 Step size 0.1
4
5 x ax
6 8 4.29981696
7 8.1 4.378930909610
8 8.2 4.459500506538
9 8.3 4.541552533785
10 8.4 4.625114267146
11 8.5 4.710213484270
12 8.6 4.796878473900
13 8.7 4.885138045273
14 8.8 4.975021537698
15 8.9 5.066558830310
Solution
Changing the subject of a formula is the same as solving an equation except that the solution is not a number but an algebraic expression (another formula).
(a) v=u+at
u+at=v Swapping sides so that u appears on the left hand side u=v−at Subtracting at from both sides to make u the subject
(b) v=u+at
u+at=v Swapping sides so that t appears on the left hand side
at=v−u Subtracting u from both sides
t= Dividing both sides by a to make t the subject
Example 12
The volume of a cone is given by the formula V= πr2h, where r is the radius of its circular
base and h is its height.
(a) Make h the subject of the formula. (b) Make r the subject of the formula.
Solution
(a) V= πr2h
πr2h=V Swapping sides so that h appears on the left hand side
πr2h= 3V Multiplying both sides by 3
h= Dividing both sides by πr2 to make h the subject
(b) V= πr2h
πr2h=V Swapping sides so that r appears on the left hand side
πr2h= 3V Multiplying both sides by 3
r2= Dividing both sides by πh
r=± Taking (positive and negative) square roots
But since r, the radius, must be a positive value, we can omit the negative square root.
r=
1. Make y the subject of each of these formulas.
(a) 2p+y= 10 (b) w= 3 +y (c) x= y− 1 (d) 2m= 5 −y (e) u= 2n+ 2y (f) x=
(g) z2=x2+y2 (h) = (i) =
(j) r= 14xy2 (k) d= 8 (l) b=c− 2ay
v–u a
---1 3
---1 3 ---1
3
---3V πr2
---1 3 ---1
3
---3V πh
---3V πh
---3V πh
---Exercise 1-06:
Changing the subject of a formula
1 4
---6y+3
a 5 --- y
2r
--- y+1
10 --- 3k
2
---2. Change the subject of each formula to the variable shown in brackets.
(a) I=Prn [P] (b) y=mx+b [m]
(c) S= [t] (d) S=V−Dn [D]
(e) d= [v] (f) V= x2h [x]
(g) v2=u2+ 2as [s] (h) v2=u2+ 2as [u]
(i) A= bh [h] (j) E=mc2 [m]
(k) E=mc2 [c] (l) tanθ= [x]
(m)d= [A] (n) s=ut+ at2 [a]
(o) K= mv2 [v] (p) v=u+at [a]
(q) z= [x] (r) T= [n]
(s) A=πr2 [r] (t) B= [m]
(u) m= [k] (v) T= 2π [l]
(w)S= (a+b+c) [a] (x) v= [r]
(y) C= 60 + 2.7k [k] (z) P= 110 + [y]
3. Change the subject of each formula to the variable shown in brackets.
(a) C= (F− 32) [F] (b) V= πr3 [r]
(c) A=π(R2−r2) [R] (d) S= 180(n− 2) [n]
(e) S= 2πr2+ 2πrh [h] (f) d= gt2 [t]
(g) B= [h] (h) D= [T]
(i) C= 12 + 8(h− 1) [h]
Listed below are 26 formulas that have been used so far in the General Mathematics course. Match each formula to its correct description below. (Some descriptions are used twice.)
1. I=Prn 2. c2=a2+b2 3. S=
4. = 5. P( ) = 1 −P(E) 6. A=πr2
7. C= 2πr 8. y=kx 9. y=mx+b
10. A= (a+b)h 11. V=Ah 12. A=P(1 +r)n
13. I=A−P 14. V= πr2h 15. A= xy
16. A=s2 17. C=πd 18. V= πr3
d t ---v2 g --- 1 3 ---1 2 ---h x ---mA 150 --- 1 2 ---1 2
---x–M s
--- n+24
8 ---m h2 ---5k 8 --- l g ---1 2
--- 2gr
y 2 ---5 9 --- 4 3 ---1 2 ---m h2 --- 6
T +1
---Group activity:
Getting the right formula
d t
---x Σx
n
--- E~
---19. A=lb 20. = 21. V= Ah
22. V=πr2h 23. A= bh
24. P(E) =
25. Percentage error= ×100%
26. m= =
Description
A. area of a trapezium B. area of a circle C. probability of a complementary event D. mean of a data set
E. volume of a prism F. volume of a cylinder
G. Pythagoras’ theorem H. circumference of a circle I. probability of an event J. volume of a cone
K. linear variation L. area of a square
M. area of a rectangle N. speed
O. area of a triangle P. gradient of a line
Q. simple interest earned R. compound interest earned
S. volume of a sphere T. area of a rhombus
U. volume of a pyramid V. equation of a line W. error of a measurement expressed as a percentage
X. final amount of an investment after compound interest x Σfx
Σf
--- 1
3
---1 2
---number of favourable outcomes total number of outcomes
---absolute error measurement
---rise run
--- vertical change in position horizontal change in position
---Study tips
Y
OUR STUDYROUTINEStart a weekly routine to develop good habits and cover all of your study commitments. For example, do more Maths on a Monday. However, don’t overplan and become obsessed with the number of hours worked each night. The quality of study is more important than the quantity. Be task-oriented rather than time-oriented.
Considerations for your study routine: How much study will I do each week?
When’s my best/worst time of day for studying? Will I alternate ‘heavy’ study days with ‘light’ ones? Do I want one night or day that is comparatively study-free? Will I study more or less on weekends?
Alternate between easy and hard tasks. Start studying straight away. Avoid time wasters such as cleaning your room, decorating title pages or even devising elaborate study timetables. Aim to develop a deep understanding of the subject rather than just memorising facts.
EQUATIONS AND FORMULAS
Sometimes when solving a problem involving a formula, the answer is not immediately found after substituting a value. Instead an equation results, which must then be solved.
Example 13
The mean of three numbers x, y and z is
M=
If three numbers have a mean of 22, and two of the numbers are 25 and 26, find the third number.
Solution
Substitute M= 22, x= 25, y= 26 into the formula to find the third number z. 22 =
66 = 25 + 26 +z 66 = 51 +z 15 =z
z= 15 The third number is 15.
Example 14
The compound interest formula
A=P(1 +r)n
calculates the amount ($A) to which a principal ($P) will grow if invested at a rate of r per annum over n years, where r must be written as a fraction or decimal. If a principal of $5000 is invested at 12.5% p.a., how many years will it take to grow to $10 000? Answer correct to 1 decimal place.
Solution
A= 10 000, P= 5000, r= 0.125. 10 000= 5000(1 + 0.125)n
10 000= 5000(1.125)n = (1.125)n
(1.125)n= 2
By guess, check and refine: n ≈ 5.9 years.
1. The formula for converting temperatures from the Fahrenheit scale to the Celsius scale is C= (F− 32)
Use the formula to convert a temperature of 38°C to °F.
2. The number of days fresh milk will keep if stored at temperature T°C is
D=
If a carton of milk lasted 4 days, at what temperature was it stored? x+ y+z
3
---25+26+z 3
---10 000 5000
---Exercise 1-07:
Equations and formulas
5 9
---3. The volume of a cylinder is
V=πr2h
where r is the radius of its base and h is its height. If a cylinder has volume 904.78 cm3
and height 8 cm, find the radius of its base to the nearest centimetre.
4. The compound interest formula
A=P(1 +r)n
shows the amount ($A) to which a principal ($P) will grow if invested at a rate of r per annum over n years, where r must be written as a decimal.
(a) What principal needs to be invested at 9% p.a. for it to grow to $4000 in 5 years? Express your answer to the nearest cent.
(b) For how many years must a principal of $4000 be invested at 9% p.a. to grow to $10 000? Answer to the nearest year.
5. The angle sum S of a polygon with n sides is S= 180(n− 2)° How many sides has the polygon with an angle sum of:
(a) 360°? (b) 1440°?
6. The formula for converting speeds from kilometres per hour to metres per second is
M= Convert 12 m/s to kilometres per hour.
7. The distance in kilometres that an observer can see to the horizon from a height of h m is
d= 8
How high (to the nearest metre) would you need to be to see a distance of 100 km?
8. The simple interest earned when a principal $P is invested at an interest rate of r per annum for n years is
I=Prn
where r must be written as a decimal. For how long must a principal of $2300 be invested at 6.5% p.a. to earn simple interest of $1196?
9. The average speed of a moving object in kilometres per hour is
S=
where d is the distance travelled in kilometres and t is the time taken in hours. (a) Find the distance travelled by a car in 3 hours if its average speed is 90.2 km/h. (b) Find the time taken (to the nearest minute) for a cyclist to travel 250 km if his
average speed is 17.5 km/h.
10. The volume of a sphere of radius r is V= πr3
Find the radius (correct to 1 decimal place) of the sphere with a volume of 2854.54 cm3.
11. Young’s rule for calculating the medicine dosage for infants is
D=
where y is the age of the infant in years and A is the adult dosage. How old is Libby if her medicine dosage is 0.6 mL and the adult dosage is 15 mL?
12. The surface area of a rectangular prism of length l, breadth b and height h is given by S= 2lb+ 2lh+ 2bh
If a rectangular prism with surface area 222 cm2 has breadth 3.5 cm and height 5 cm,
find its length.
5k 18
---h 5
---d t
---4 3
---13. Washing removes 20% of a deep stain at each wash. After further washes, the percentage of the original stain left is given by the formula
A= 100(0.8)w
where w is the number of washes.
(a) What percentage of the stain remains after 2 washes? (b) Show that 3 washes remove about half of the original stain.
(c) At least how many washes are needed to reduce the stain to less than 5% of its original content?
14. The formula for converting Australian dollars ($A) to New Zealand dollars ($NZ) is NZ = 1.2831A
Convert the following $NZ amounts to $A, correct to the nearest cent.
(a) $NZ20 (b) $NZ38 (c) $NZ100
15. The number of times per minute a cricket chirps on a hot summer’s night is n= 8T− 24
where T is the temperature in °C. What is the temperature if the cricket chirps 144 times per minute?
16. The area of an annulus is
A=π(R2−r2)
where R is the longer radius and r is the shorter radius. Calculate the shorter radius of this annulus (to the nearest centimetre) if its area is 794.82 cm2.
17. The maximum distance d m that a ball covers if thrown with velocity v m/s is
d=
where g= 9.8 m/s2 is the gravitational acceleration. At what velocity was a ball thrown
if it covered a distance of 25.8 m? Answer correct to 1 decimal place.
18. The body mass index (BMI) of an adult is B=
where m is the mass in kilograms and h is the height in metres.
(a) Kate is 1.76 m tall and has a BMI of 24.2. Calculate her weight to the nearest kg. (b) Stefan weighs 81 kg and has a BMI of 24.8. Calculate his height to the nearest cm.
19. Given that E=mc2, find c if m= 0.05 and E= 4.5 × 1015.
20. The time (in seconds) it takes a swing to move back and forth once is
T= 2π
where l is the length of the swing (in metres) and g= 9.8 m/s2 is the gravitational
acceleration. Find the length of the swing (correct to 2 decimal places) if it takes 3.17 seconds to move back and forth once.
21. Use the compound interest formula
A=P(1 +r)n
to determine the number of years it will take a principal to double in value if invested at 5% p.a. Answer to the nearest 0.1 year. Hint: Let A= 2P.
17 cm
r
v2
g
---m h2
---22. The surface area of a cone is
S=πr(r+s)
where r is the radius of the base and s is the slant height of the cone. Calculate the slant height of the cone (correct to 2 decimal places) that has a base radius of 3.5 cm and surface area of 101.16 cm2.
23. The blood alcohol content (BAC) of a person who consumes alcoholic drinks is BAC = 0.0012na
where n is the number of drinks and a is the amount of alcohol in each drink. If Mark drank 4 glasses of wine (each 150 mL) and registered a BAC level of 0.0792:
(a) how many millilitres of alcohol were present in each glass? (b) what percentage of the wine was alcohol?
24. The population density of a country (in persons per square kilometre) is given by the formula
D=
where P is the population and A is the area in square kilometres. Calculate the population of China correct to 3 significant figures if its area is 9.57 × 106 km2 and its population
density is 248 persons/km2.
25. The formula for calculating Samantha’s income tax T is T= 11 380 + 0.42(I− 50 000)
where I is her taxable income. Calculate Samantha’s taxable income if her income tax was $13 102.
There are numerous models for prescribing medicine to infants. Two of them are Fried’s rule and Young’s rule.
Fried’s rule is D= , where m is the age of the infant in months and A is the adult dosage.
Young’s rule is D= , where y is the age of the infant in years and A is the adult dosage.
Create a spreadsheet like the one on the next page that investigates and compares the two models. Enter the adult dosage and the infant’s age in months, then use the spreadsheet to calculate: the age of the infant in years
the infant dosage using both rules the difference between the two values.
s
r
P A
---Spreadsheet modelling activity:
Two models for infant medicine
mA 150
---1. Is there an age value for which the infant dosages given by both rules are identical?
2. When do the two rules differ the most?
LINEAR FUNCTIONS
A function like y=−2x+ 10 is an algebraic relationship between two variables. It can be thought of as a ‘number machine’ that converts an input value, x, into an output value, y.
6 FUNCTION −2
x y=−2x+ 10 y
Independent variable Dependent variable
Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable. y=−2x+ 10 is called a linear function because it involves a linear expression (−2x+ 10) and its graph on the number plane is a straight line.
A B C D E
1 INFANT DOSAGE CALCULATOR
2
3 Adult dosage (mL) 15
4 Age of infant (months) 30
5
6 AGE OF INFANT DOSAGE (mL)
7 Months Years Fried’s rule Young’s rule Difference
8 30 2.50 3.00 2.59 0.4138
9 31 2.58 3.10 2.66 0.4429
10 32 2.67 3.20 2.73 0.4727
11 33 2.75 3.30 2.80 0.5034
12 34 2.83 3.40 2.87 0.5348
13 35 2.92 3.50 2.93 0.5670
14 36 3.00 3.60 3.00 0.6000
15 37 3.08 3.70 3.07 0.6337
16 38 3.17 3.80 3.13 0.6681
17 39 3.25 3.90 3.20 0.7033
18 40 3.33 4.00 3.26 0.7391
Input Output
A linear function has the form y=mx+b.
The gradient, m, is the rate of change of y relative to x.
The y-intercept or vertical intercept, b, represents the value of y when x is zero.
The graph of y=mx+b is a straight line, demonstrating that y changes at a steady rate.
m= =
m= =
change in y
change in x
--- change in dependent variable
change in independent variable
---rise run
--- vertical change in position
horizontal change in position
---Functions involving higher powers of x such as x2 and x3 are called non-linear functions
because their graphs are not lines but curves. Examples include y= 3x2−x+ 1, a quadratic
function (whose graph is a parabola), and y=−2x3+ x+ 8, a cubic function. These
non-linear functions will be examined in Chapter 9.
Linear models
When scientists and researchers observe number patterns occurring in nature and society, they try to find or fit a mathematical formula to represent the relationship. This is called algebraic modelling. A model approximates real life phenomena algebraically and can be described using a formula, a table of values or a number plane graph.
If the observed number pattern suggests a linear relationship, we use the linear function y=mx+b to model the situation. This is called a linear model.
Example 15
A criminologist studying crime in a major city found a linear relationship between P, the number of police patrolling the city, and C, the number of crimes committed per week. Some of her results are illustrated in the table.
(a) What is the independent variable?
(b) Find the linear function of the form C=mP+b. (c) What is the gradient and what does it represent?
(d) What is the vertical intercept and what does it represent?
(e) Use the function to predict the number of crimes per week when 400 police are on patrol.
Solution
(a) P, the number of police on patrol, is the independent variable.
(b) m =
= Choosing (50, 3100) and (150, 2800)
= =−3
∴C =−3P+b
To find b, substitute another point, say (300, 2350). 2350=−3(300) +b
2350=−900 +b b= 3250
∴C=−3P+ 3250
(c) The gradient −3 represents the reduction in the number of crimes for each new police officer added. As P increases by 1, C decreases by 3. According to this linear model, for every new police officer added, the number of crimes decreases by 3.
(d) The vertical intercept 3250 represents the number of crimes committed per week if P= 0 (i.e. no police on patrol).
(e) When P= 400
C=−3(400) + 3250
= 2050 crimes per week
P (no. of police) 50 150 250 300
C (crimes per week) 3100 2800 2500 2350
1 2
---change in C change in P
---2800–3100 150–50
---300 –
---Line of best fit
Real observed data, also called empirical or experimental data, usually do not fit the linear relationship y=mx+b so strictly or reliably. There are two reasons for this:
There are always errors and inaccuracies associated with the measurement of data. Often, the patterns and relationships in real life phenomena are actually more complicated
than a simple linear formula.
In these cases, we use a linear model to approximate the pattern because it is simpler. We introduce the linear model by drawing a line of best fit through the graphed empirical data.
Example 16
This table shows the progressive score (total runs) of a cricket team playing in a one-day 50-over match. (An over is a bowler’s round of six balls bowled.)
(a) Graph this data and construct a line of best fit. (b) Find the equation of the line.
(c) Use the equation to estimate the final score after 50 overs.
(d) Use the line to estimate the over in which the team’s score reached 100.
(e) A cricket team’s run rate is the average number of runs scored per over. According to the linear model, what was this team’s run rate?
(f) What are the limitations of this linear model?
Solution
(a)
n (overs) 7 1 7 20 25 33 45
S (runs) 60 87 105 123 144 172 P
5 1 0 1 5 20 25 30 35 40 45 50 180
160
140
120
100
80
60
40
20
0
Runs,
S
Overs, n
Progressive score of a cricket team in a 50-over match
Use a transparent ruler when constructing a line of best fit.
Your line may differ slightly from this one.
(20, 105)
(b) The equation has the form S=mn+b.
Choosing two points on the line, (20, 105), (30, 132),
m=
= =
= 2.7
∴S= 2.7n+b Substitute (20, 105) to find b.
105= 2.7(20) +b = 54 +b b= 51
∴S= 2.7n+ 51 (c) When n= 50 S= 2.7(50) + 51
= 186 runs
(d) The score reached 100 in the 19th over.
(e) The team’s run rate is the gradient of the line. Run rate = 2.7 runs/over.
(f) The vertical intercept of 51 implies that the team already had a score of 51 when the game started (0 overs). This is not possible. Also, a straight line graph suggests that the team’s score increases at a steady rate, but in reality the players all have different batting abilities and would not score at the same rate.
1. Of the functions listed below, which ones are:
(a) linear? (b) quadratic? (c) cubic? (d) none of these
A. y= –3x2 B. y= –4x – 1
C. y=x2 – 2x+ 1D. y= –x3+x+ 4
E. y= 12 – x F. y=x3 – x2
G. y= 2x2+ 9 H. y= 2x
I. y= – J. y= x
K. y= x3+ 6x2+ 7x – 10 L. y= 2. Find the linear function for each of these tables of values.
(a) x 4 7 1 2 20 (b) k 6 1 0 1 3 1 5
y 10 31 66 122 C 152 120 96 80
change in S change in n
---132–105 30–20
--- 27 (20, 105) is actually one of the observed data points. 10
---From the graph, a vertical intercept of 51 looks reasonable.
From the graph, a final score of 186 looks reasonable.
A line of best fit:
represents most or all of the points as closely as possible
goes through as many points as possible
has roughly half of the points above it and roughly half of the points below it
is drawn so that the distance between each point and the line is kept at a minimum.
Exercise 1-08:
Linear functions
1 x
--- 1
4
---1 2
--- x–7
---3. The table below shows the cost of water usage for different volumes of water.
(a) Find a linear formula for C in terms of v. (b) What does the gradient represent?
(c) What does the vertical intercept represent?
(d) What is the independent variable and what does it represent?
4. A road safety analyst studied the relationship between the populations of towns and the road accident rate (per day) in those towns.
(a) Graph this data and construct a line of best fit. (b) Find the equation of the line.
(c) What is the daily road accident rate per 1000 population?
(d) What is the daily road accident rate when the population is 24 000? (e) Estimate the population for which the daily road accident rate is 20.
5. This conversion graph shows the linear relationship between the Celsius and Fahrenheit scales for measuring temperature.
Volume, v (kL) 8 1 5 20 26 38 50
Cost, C ($) 98.8 104.75 109 114.1 124.3 134.5
Population, P (thousands) 1 2 1 7 20 32 45 50 61 67
Accidents per day, A 2 3 4 7 1 0 1 1 1 3 1 5
5 1 0 1 5 20 25 30 35 40 45 50 90
80
70
60
50
40
30
0
Fahrenheit scale,
F
(°F)
Celsius scale, C (°C)
100 110
Celsius–Fahrenheit temperature conversion graph
120
(20, 68)
(a) Find the formula for F in terms of C. (b) What is the independent variable?
(c) What is the vertical intercept and what does it represent? (d) Use the formula to convert:
(i) 12°C to °F (ii) 86°F to °C (e) Use the graph to convert:
(i) 12°C to °F (ii) 86°F to °C
6. Lizzie recorded the value of her computer every 6 months. She graphed this data and drew a line of best fit as shown below.
(a) Find the equation of Lizzie’s line of best fit.
(b) Use the equation to calculate the original value of her computer. (c) What is the gradient of the line and what does it represent? (d) Calculate the value of her computer after:
(i) 20 months (ii) 3 years
(e) When will the value of the computer be zero? Answer in years and months.
Months after purchase, m 6 1 2 1 8 24 30 36
Value of computer, $V 2900 2700 2510 1850 1200 750
1 2
---6 1 2 1 8 24 30 36 42 48
3600
3200
2800
2400
2000
1600
1200
800
400
0
Value of computer,
V
($)
Months after purchase, m
4000
7. The length of a shoe (in inches) has a linear relationship with its shoe size.
(a) What is the independent variable?
(b) Find the linear function for L in terms of S. (c) What does the gradient represent?
(d) What does the vertical intercept represent? (e) What is the length of a shoe of:
(i) size 6? (ii) size 7 ? (f) What size is a shoe of length:
(i) 8 inches? (ii) 13 inches?
8. In a factory, the weekly cost $C of manufacturing palm-sized computers is shown in the table, where p is the number of units produced.
(a) What is the dependent variable? (b) Find the linear function.
(c) What does the gradient represent?
(d) What does the vertical intercept represent?
(e) What is the cost of producing 200 palm-sized computers? (f) How many units can be produced for $60 000?
9. Erin observed the following relationship between the volume of petrol V (in litres) in her car and the distance d (in kilometres) travelled on this volume.
(a) Graph this data and construct a line of best fit. (b) Find the equation of this line.
(c) Use the equation to predict:
(i) the distance travelled on 45 L of petrol
(ii) the number of litres of petrol needed to travel 350 km (d) What is the gradient, m, and what does it represent?
(e) Use the formula F= to calculate the fuel consumption rate of Erin’s car in litres per 100 km correct to 1 decimal place.
S (size) 2 5 7 8 1 2
L (inches) 9 1 0 1 0 1 1 1 2
p (units) 50 85 120 136 157
C ($) 21 900 31 350 40 800 45 120 50 790
V (L) 1 8 22 28 38 40 56
d (km) 179 225 301 387 412 580
2 3
--- 1
3
---1 2
---2 3
---Programmable, statistical and graphics calculators usually have a feature that allows you to input the (x, y) coordinates of a set of points, then fits a straight line of the form y=mx+b through them, such that the distances between the points (observed data) and the line (linear model) are minimised. This is called a linear regression or line of best fit function.
Use the linear regression function of your calculator to redo Example 16 or question 9 in the previous exercise. Enter the coordinates of the set of observed data and see how the calculator generates a line of given gradient and vertical intercept. Compare the calculator’s linear model to your own.
Equipment: Tape measure or height chart, graph paper, graphing software or graphics calculator (if available).
Is it possible to predict a person’s shoe size from their height?
1. Select a sample of about 20 people. Measure their heights, h, to the nearest centimetre and record their shoe sizes, s.
2. Plot the data on a graph with s on the vertical axis (dependent variable).
3. Construct a line of bes
