warrnambool college maths methods unit 1 & 2...

Warrnambool College Maths Methods

Unit 1 & 2 2014

Unit 1 Topics:

• Linear functions

• Quadratic functions

• Cubic functions

• Exponential & logarithmic functions

• Functions & relations

Unit 2 Topics:

• Circular functions

• Differentiation

• Matrices

• Probability

• Combinatorics

Assessment tasksThere will be a test for each of these topics. Students are allowed to bring their workbooks into

the test (with notes & questions). CAS calculators can be used in most tests, although some will

have a calculator free component. Tests will contribute 80% of the mark for each topic area.

20% of the mark for each topic will come from having completed the set Khan Academy tasks.

Each semester there will be one problem solving / project task and exam.

Semester reports will include a grade for each topic test, as well as the semester exam and

problem solving task..

At Warrnambool College students will have met the outcomes for a unit if they:

• achieve 40% or better on all topics.

• complete the set class work and completed the relevant revision work.

• complete the problem solving / project task.

• attend at least 80% of classes.

Maths Methods Blog PageTo give students the opportunity to learn the concepts and work through at their own pace,

podcast videos and printed notes for all topic areas have been prepared. These are all available

on the school network and the podcast videos can be accessed freely through the iTunes store

(Maths Methods Podcast: itpc://mathsmethods.podomatic.com/rss2.xml) or the blog site.

The blog site http://methods.global2.vic.edu.au/ is also available for students to download these

materials, as well as exam revision, worksheets, Khan Academy links etc. All students will be

given a username & password to contribute to the site. Students can also post questions and

answer other students questions.

It is expected that all students will have obtained these before resources in advance of the topic

and are watching / reading as part of a regular homework routine. This preparation will allow

students to better use the class time to work on problems and clarify any difficulties.

Khan AcademyThe Khan Academy website provides video tutorials and exercises in all areas of mathematics. All

students will be able to sign into the Khan Academy using their Warrnambool College GMail

account. (Please make your username your full name for identification.) Class teachers

([email protected] or [email protected] ) need to be added as

coaches.

CAS calculatorsIt is presumed that all students will have a CAS calculator for class and assessment tasks. Some

of the assessment tasks will require the use of the calculator, for other tasks it will be of

assistance. The essential skills that students will need for the start of the year include:

• Key operations

• Saving documents

• Navigating through multiple tabs

• Drawing graphs

• Finding values from equations

• Solving equations

Text bookIt is expected that all students will have a current (third edition) or the previous edition of the text

book: MathsQuest 11 Mathematical Methods CAS and separate CAS calculator guide. (These

have blue covers). Older (brown covered) editions are not advised.

http://itpc://mathsmethods.podomatic.com/rss2.xml

http://itpc://mathsmethods.podomatic.com/rss2.xml

http://methods.global2.vic.edu.au

http://methods.global2.vic.edu.au

mailto:[email protected]




CAS introductory task

For the linear function y =10 + 2x5 :

a) Plot the graph over the domain (-20,20).

b) Find the value of y when x = 12.

c) Find the y and x intercepts.

d) Find the value of x for which y = 12.

e) Find the point where the function intersections the line y = 6 − 2x .

Show how questions can be completed using the CAS calculator. You must show at least two

ways that questions b - e can be answered using the calculator.

Save this document on your calculator for all of your linear functions work.

Headstart work tasks

1A - Solving linear equations: Q1 – 3 (ESQ)

1B - Rearrangement and substitution: Q2, 4

1C - Gradient of a straight line: Q1–6 (ESQ), 7,8

1D - Equations of the form y = mx+c: Q1 – 5 (CAS activity), 6 - 8 (ESQ)

5Chapter 1 Linear functions

b 1 Write the inequations. b 3m 5 72 Subtract 5 from both sides. 3m 123 Divide both sides by 3. Reverse the

inequality sign, as you are dividing by a negative number.

m 4

Method 2: Technology-enabled

a,b 1

2 Write the answers. a Solving 6x 7 ! 3x 5 for x, gives x ! 4.

b Solving 3m 5 7 for m, gives m 4.

Linear equations can be solved by rearranging to make the pronumeral the subject.1. When solving linear inequalities, imagine an equals sign in place of the inequality sign 2. and solve as if it were a linear equation. Remember to keep writing the original inequality sign at each step, but reverse the sign if dividing or multiplying by a negative number.

REMEMBER

Solving linear equations and inequations1 WE1 Solve the following linear equations.

a 3x 19 13 b 4x 25 7 c 9x 19 2

d 3 14

5x e 12 3

35

x f 4 63

7 3x

g 7 43

8 9x

h 235

1 10x i 7 8

477

x

2 WE2 Solve the following linear equations.

a 2x 9 3(2x 11) b 7x 1 17(3x 13) c x 11 2(x 12)

d 3x 7 3(35 2x) e x x26

53

f x x x113

2 149

( )

EXERCISE

1A

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Equation solvers

To solve each of these inequations using a CAS calculator, complete the following steps. On a Calculator page, press:

MENU b3

1Complete the entry lines as:solve(6x 7 3x 5, x)solve( 3m 5 7, m)Press ENTER · after each entry.

6

g 4 663

13 34

x x h x x109

2 7 35

( ) i 6 75

5 14

1x x

j 2 293

448

2x x k 7 9

921

318

x x l 172

49 25

5x x

3 WE3 Solve the following linear inequations.

a 15 6x 2 b 11 37

6x c 19

43 1

x

d 16 4x 7(1 x) e 3x 7 2(35 2x) f 43

2 16

x x( )

Rearrangement and substitutionWhen there is more than one pronumeral involved in an equation, we may rearrange the equation to make a particular pronumeral the subject using the same rules of equation solving described in the previous section.

WORKED EXAMPLE 4

Rearrange each of the following to make the pronumeral in parentheses the subject.

a 6x 8y 48 0 (y) b s ut at k12

2 (u) c Tmkk2 (k)

THINK WRITE/DISPLAY

a 1 Write the equation. a 6x 8y 48 02 Add 48 to both sides. 6x 8y 483 Subtract 6x from both sides. 8y 48 6x

4 Divide both sides by 8. yx

yx

or y x

48 68

24 34

244

34

5 Cancel if possible. Here, divide the numerator and denominator by 2.

yx24 3

4

6 Other ways of representing the answer are shown opposite.

or y x

x

x

244

34

634

34

6

b 1 Write the equation. b s ut at

s at ut

s at ut

s att

u

u

12

12

2 2

22

2

2

2

2

2

ss att

2

2

2 Subtract 12

2at from both sides. s at ut12

2

3 Multiply both sides by 2. 2 22s at ut

4 Divide both sides by 2t. 22

2s att

u

5 Write the equation with the desired pronumeral on the left.

us at

t2

2

2

1B

Maths Quest 11 Mathematical Methods CAS for the TI-Nspire


4 Reverse so that s is given on the left.

v u as

v u as

v ua

s

sv u

a

2 2

2 2

2 2

2 2

2

2

2

2

c 1 Match the pronumerals with the given information.

c u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

2 Write the formula that has s as the subject (see part b above).

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

3 Substitute the values given in step 1.

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

4 Simplify and evaluate.

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

5 Explain the answer in words. The object travels 42 m in its initial direction.

Equations may be rearranged by applying the same rules as those used to solve equations.

REMEMBER

Rearrangement and substitution 1 WE4 Each of the following is a real equation used in business, mathematics, physics or

another area of science. Make the pronumeral shown in parentheses the subject in each case.a A L P (P) b A lw (l)

c vdt

(t) d C 2 r (r)

e E ( ) f FkQqr2

(r)

g Fd mv mu12

12

2 2 (v) h v rT ( )

i S 2w(l h) 2lh (w) j S 2 r 2 rH (H)

2 WE5 Calculate the value of the subject (the first mentioned pronumeral), given the values of the other pronumerals.

a Ik

d 2 k 60, d 15

b E K mgh K 250, m 2, g 10, h 5

c D n( )12 n 3, 2.8

d E hf0 W h 6.62, f0 5000, W 20 000

e v r y2 2 2, r 1.6, y 1

EXERCISE

1B


4 Reverse so that s is given on the left.

v u as

v u as

v ua

s

sv u

a

2 2

2 2

2 2

2 2

2

2

2

2

c 1 Match the pronumerals with the given information.

c u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

2 Write the formula that has s as the subject (see part b above).

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

3 Substitute the values given in step 1.

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

4 Simplify and evaluate.

u

v

a

sv u

a

10

4

1

24 102 1

16 1002

842

2 2

2 2

442

5 Explain the answer in words. The object travels 42 m in its initial direction.

Equations may be rearranged by applying the same rules as those used to solve equations.

REMEMBER

Rearrangement and substitution 1 WE4 Each of the following is a real equation used in business, mathematics, physics or

another area of science. Make the pronumeral shown in parentheses the subject in each case.a A L P (P) b A lw (l)

c vdt

(t) d C 2 r (r)

e E ( ) f FkQqr2

(r)

g Fd mv mu12

12

2 2 (v) h v rT ( )

i S 2w(l h) 2lh (w) j S 2 r 2 rH (H)

2 WE5 Calculate the value of the subject (the first mentioned pronumeral), given the values of the other pronumerals.

a Ik

d 2 k 60, d 15

b E K mgh K 250, m 2, g 10, h 5

c D n( )12 n 3, 2.8

d E hf0 W h 6.62, f0 5000, W 20 000

e v r y2 2 2, r 1.6, y 1

EXERCISE

1B

10

3 Make the pronumeral in parentheses the subject, and find its value using the given information.

a A l (l) A 60

b V r43

3 (r) V 1000

c v u at (a) v 25, u 0, t 6

d Tlg

2 (l) T 4, g 9.8

e Kc 2

1 (c) K 6.9, 0.05

4 WE6 The perimeter, P, of a rectangle of length l and width w may be found using the equation P 2(l w).a Find the perimeter of a rectangle of length 16 cm and width 5 cm.b Rearrange the equation to make w the subject.c Find the width of a rectangle that has perimeter 560 mm and length 240 mm.

5 The area of a trapezium (figure A) is given by Aa b

h2

, where

Figure A

a

h

b

AreaA

a and b are the lengths of the parallel sides, and h is the height.a Find the area of the trapezium shown in figure B.

b Using figure A, find an equation for determining side a in terms of the area A and side b.

c Find a in figure C.

6 The size of a 2-year investment account with a particular bank is given by

A Dr

1100

2

, where A is the amount ($) in the account after two years, D is the initial

deposit ($) and r is the interest rate (%).a Find the amount in such an account after two years if the initial deposit was $1000 and

the interest rate was 6%.b Make r the subject of the equation.c Find the rate required for an initial deposit of $1000 to grow to $2000 after 2 years.

7 The object and image positions for a lens of focal length f are related by the formula1 1 1u v f

, where u is the distance of the object from the lens

and v is the distance of the image from the lens.a Make f the subject of the equation.b Make u the subject of the equation.c How far from the lens is the image when an object is

30 cm in front of a lens of focal length 25 cm?

8 The length of a side of a right-angled triangle can be found using Pythagoras’ theorem: c a b , where c is the length of the

Figure B

9 m

16 m

21 m

62 cm50 cm

a

Area = 2000 cm2

Figure C

32

b

4



THINK WRITE

a Since the angle the line makes with the positive x-axis is given, the formula m tan ( ) can be used.

a m tan ( ) tan (40 )

correct to 3 decimal places

b 1 The angle given is not the one between the graph and the positive direction of the x-axis. Calculate the required angle .

b 180 60 120

2 Use m tan ( ) to calculate m to 2 decimal places.

m tan ( ) tan (120 )

1.73

The gradient (m) of a straight line may be calculated using the following formulas.

m riserun

y

x

rise

run

my yx x

2 1

2 1, where (x1, y1) and (x2, y2) are points on the line

m tan ( ), where is the angle the line makes with the positive direction of the x-axis

REMEMBER

Gradient of a straight line 1 WE7 Calculate the gradient of each of the following linear graphs using the intercepts

shown.a

6

3

y

x

b

2

6

y

x

2 WE8 Without drawing a graph, calculate the gradient of the line passing through:

a (2, 4) and (10, 20) b (4, 4) and (6, 14)c (10, 4) and (3, 32) d (5, 31) and ( 7, 25)

y

x60

y

x60

EXERCISE

1C

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14

3 WE9a Calculate the gradient (accurate to 3 decimal places) of a line making the angle given with the positive x-axis.a 50 b 72c 10 d 30e 150 f 0g 45 h 89

4 WE9b Calculate the gradient of each line below. Give answers to 2 decimal places.

a y

x43

b y

x69

c y

x28

d y

x15

5 Which of these lines has:a a non-zero positive gradient?b a negative gradient?c a zero gradient?d an undefined gradient?

6 MC

a Which of the following lines has a gradient of 2?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

yA C

E

B D

b Which of the following lines has a gradient of 3?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

yAC

E

B

D

7 Burghar plots the coordinates of a proposed driveway on a plan which is shown at the top of the facing page. What is the gradient of the proposed driveway?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

y

A

C

B D


14

3 WE9a Calculate the gradient (accurate to 3 decimal places) of a line making the angle given with the positive x-axis.a 50 b 72c 10 d 30e 150 f 0g 45 h 89

4 WE9b Calculate the gradient of each line below. Give answers to 2 decimal places.

a y

x43

b y

x69

c y

x28

d y

x15

5 Which of these lines has:a a non-zero positive gradient?b a negative gradient?c a zero gradient?d an undefined gradient?

6 MC

a Which of the following lines has a gradient of 2?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

yA C

E

B D

b Which of the following lines has a gradient of 3?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

yAC

E

B

D

7 Burghar plots the coordinates of a proposed driveway on a plan which is shown at the top of the facing page. What is the gradient of the proposed driveway?

54321

12345

5 4 3 2 1 1 2 3 4 50 x

y

A

C

B D



Driveway

Garage

2 m

17 m

8 An assembly line is pictured below. What is the gradient of the sloping section? (Give your answer as a fraction.)

0.85 m

15 m

BOFFOMade inAustraliaBOFFOMade inAustralia

BOFFOMade inAustralia

9 Determine the value of a in each case so the gradient joining the points is equal to the value given.a (3, 0) and (5, a) gradient: 2b (2, 1) and (8, a) gradient: 5c (0, 4) and (a, 11) gradient: 3d (a, 5) and (5, 1) gradient: 2

10 For safety considerations, wheelchair ramps are constructed under regulated specifications. One regulation requires that the maximum gradient of a ramp exceeding 1200 mm in length is 1

14.

a Does a ramp 25 cm high with a horizontal length of 210 cm meet the requirements?b Does a ramp with gradient 1

18 meet the specifi cations?

c A 16 cm high ramp needs to be built. Find the horizontal length of the ramp required to meet the specifi cations.

Equations and functions and the CAS calculatorIn previous years you studied the graphs of equations such as y 2x 1. This equation describes the relationship between x and y. In other words, the y-values can be determined from the x-values. This means that ‘y is a function of x’, which is abbreviated to y f (x). So the rule y 2x 1 can also be written as f (x) 2x 1.

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Using a gradient to fi nd

the value of a parameter


4 Plot the points (0, 0) and (1, 43) on a

set of axes. Note that 43 is 11

3, which is

a little less than 112

.

y

x

4–3(1, )

(0, 0)1

1

2

The general equation for a straight line of gradient 1. m and y-intercept c is y mx c.Lines with the same gradient (2. m) are parallel.To sketch a linear graph:3. (a) Let x 0 and fi nd the y-intercept.(b) Let y 0 and fi nd the x-intercept.(c) If (0, 0) is an intercept, fi nd another point on the line by substituting x 1

(or any other convenient non-zero value).(d) Mark and join the intercepts or the intercept at the origin and the second point with

a straight line.

REMEMBER

Sketching linear functions 1 Use a CAS calculator or other method to sketch graphs of the following equations on the same

set of axes.a y x b y 2x c y 3x d y x e y 2x

2 What is the effect on the graph of the number in front of the x (the ‘x-coefficient’ or ‘gradient’)?

3 Use a CAS calculator or other method to sketch graphs of the following equations on the same set of axes.a y x 1 b y x 2 c y x 3 d y x 4

4 Use a CAS calculator or other method to sketch graphs of the following equations on the same set of axes.a y 2x 1 b y 2x 7 c y 3x 6 d y 3x 5

5 What is the effect on the graph of the number at the end of the equation (the ‘y-intercept’)?

6 Write the equation of a line having the following properties (where m gradient and c y-intercept).a m 2, c 7 b m 3, c 1 c m 5, c 2d m 2

3, c 1

3 e m 34 , c 1

2f y-intercept 12, gradient 2

7 Rearrange the following equations and state the gradient and y-intercept for each.a 2y 8x 10 b 3y 12x 24 c y 3x 1d 16 4y 8x e 21x 3y 27 f 10x 5y 25g 11y 2x 66 h 8x 3y 2 0 i 15 6y x 0j 2y 7 5x 0

EXERCISE

1D

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Linear graphs

20

8 State the equation for each of the following graphs.

a 54321

12

1

6

1 20 x

y

1–2

b 321

12345

2 20 x

y

c 321

12345

12 1 20 x

y d 54321

12345

1 1 2 3 4 5 60 x

y

e 54321

12345

1 1 2 3 40 x

y f 987654321

123456789

3 2 1 1 2 30 x

y

9 WE 10 Sketch graphs of the following linear equations, showing x- and y-intercepts.a y 6x 18 b y 3x 21c y 2x 3 d y 10 5xe y 9x 30 f y 2(x 8)

10 WE 11 Sketch graphs for each of the following equations.a 2x 3y 6 b 4x 5y 20c 6x 3y 18 d 7x 5y 35

11 Sketch the graph for each equation.a 6x 7y 42 0 b 5x 2y 20 0c 3x 4y 16 0 d y 3x 6 0

12 WE 12 Sketch the graph for each equation.a x y 0 b x y 0c 2x y 0


580 Answers

Answers

CHAPTER 1Linear functionsExercise 1A — Solving linear equations and inequations 1 a 2 b 8 c 7

3 d 7

e 1 f g h i

2 a 6 b 5 c 13 d 112

9

e 12 f 5 g 9 h 7659

i 3 j 4 k 9 l 7

3 a x 136

b x 313

c x 3 d x 3

e x 11 f x 32

Exercise 1B — Rearrangement and substitution

1 a P A L b lAw

c tdv

d rC2

e E

2 f rkQqF

g vFd mu

m2 2

h vrT

2

i wS lhl h

22( ) j H

S rr

22

2

2 a 0.267 b 350 c 7 d 13 100 e 2.498

3 a l A, .7 746 b rV3

46 2043 , .

c av ul

, .4 167 d l gT2

3 9722

, .

e cK( )

,1

26222

4 a 42 cm

b wP

l wP l

22

2or c 40 mm

5 a 240 m2

b aAh

b aA bhh

2 2or c 18 cm

6 a $1123.60

b rAD

A D

D100 1 100 c 41.4%

7 a fuvu v

b uv ff v

c 150 cm

8 b 2 9 h25

cm

Exercise 1C — Gradient of a straight line 1 a 2 b 1

3

2 a 2 b 5 c 4 d 12

3 a 1.192 b 3.078 c 0.176 d 0.577 e 0.577 f 0 g 1 h 57.290 4 a 0.93 b 2.61 c 0.53 d 3.73 5 a D b C c A d B 6 a B b E 7 2

17 8 17

300 9 a 4 b 31 c 5 d 3 10 a No b Yes c 224 cmExercise 1D — Sketching linear functions 1 a–e

x

y

y x

y 2xy 2x

y x

y 3x

2 The higher the number, the steeper the graph. Positive values make the graph slope up when moving (or tracing) to the right; negative values make the graph slope down when moving to the right.

3 a–d

x

1

4

y

4

y x 4

y x 1

y x 2

y x 3

4 a–d

x0

1

6

5

7

y

11

y 2x + 1

y 2x 7

y 3x 5

y 3x 6

5 The number is where the graph cuts the y-axis (hence the name ‘y-intercept’).

6 a y 2x 7 b y 3x 1 c y 5x 2 d y x2

313 e y x3

412 f y 2x 12

7 a 4, 5 b 4, 8 c 3, 1 d 2, 4 e 7, 9 f 2, 5 g

211

, 6 h 83

23

,

i 16

52

, j 52

72

,

581Answers

Answ

ers 1A

1E

8 a y 4x 2 b y 3x 5 c y x65

2

d y 56x 5 e y 2x 1 f y 5x

9 a y

x3

18

b y

x

21

7

c y

x

3

3 – 2

d y

x

10

2

e y

x

30

10 — 3

f y

x8

16

10 a y

x

2

3

b y

x

4

5

c y

x3

6

d y

x

7

5

11 a y

x

6

7

b y

x

10

4

c y

x

4

16 — 3

d y

x2

6

12 a y

x

(1, 1)

b y

x

(1, 1)

c y

x

(1, 2)

13 D 14 E 15 AExercise 1E — Simultaneous equations 1 a

x

1

1

y

11

y 4x

y 2x

b

x

20

5

y

55

y 20

y 3x 5

253( ), 20

c

x

y

9 ,4

474( )

y 3x 5

y 7x 4

d

x

5

0

y

( 3, 3)

y 4 3xy 6x 5

e

x

5

5

y

5

y 10x 1

y 6 2x712

296( ),

f

x

14

yy 17 9x

y 14 x

38

1098

( )

179

,

17

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