1. Linear functions Home Page: y a x h k 122 SAC Analysis ... · PDF file1. Linear functions Home Page: ... 1.6 Linear modelling 25 ... This is the basic quartic graph. It has one

iii

Introduction

1. Linear functions

Home Page:

Can the 13-minute barrier be broken? 1

Prep Zone

2

Replay File

21.1 Solving linear

equations 31.2 The gradient of a line 71.3 The equation of a straight line 121.4 Sketching linear graphs 171.5 Simultaneous linear equations 201.6 Linear modelling 251.7 Relations and functions 301.8 Coordinate geometry 37

SAC Analysis Task:

Item response 43

SAC Application Task:

Yes—making it easy 44

Chapter Review

45

2. Quadratic functions

Home Page:

Soft landings 49

Prep Zone

50

Replay File

502.1 Polynomials 512.2 Expanding

expressions 532.3 Factorising 572.4 Factorising quadratics by rule 602.5 Factorising by completing the square 622.6 The null factor law 652.7 Solving quadratics over

R

692.8 The quadratic formula 712.9 Using the discriminant 732.10 Translations of parabolas 772.11 Dilations of parabolas 812.12 The intercepts of a parabola 852.13 Solving equations from graphs 892.14 Simultaneous linear and quadratic equations 91

SAC Analysis Task:

Investigation 95


Too much, too little 96

Chapter Review

97

3. Cubic functions

Home Page:

You can’t keep a good idea down 101

Prep Zone

102

Replay File

1023.1 Expanding cubics 1033.2 Long division of

polynomials 1043.3 The remainder theorem and factor theorem 1073.4 Sum and difference of cubes 110

3.5 Solving cubic equations 1113.6 Sketching graphs of cubic functions 1153.7 Graphing cubics of the form

y

=

a

(

x

–

h

)

3

+

k

1223.8 Domain and range of cubic functions 1283.9 Polynomial models 1313.10 Transposition 138

SAC Analysis Task:

Assignment 141


Graphs of the polynomial

y

=

a

(

x

–

h

)

n

+

k

,

n

> 3 142

Chapter Review

143

4. Advanced functions and relations

Home Page:

When is a knot not a knot? 147

Prep Zone

148

Replay File

1484.1 Quartics 1494.2 Solving polynomial

equations (including to degree 4) 1544.3 Hyperbola and truncus 1574.4 The square root function 1624.5 Circles 1654.6 Inverse functions 1684.7 Absolute value graphs 173

SAC Analysis Task:

Assignment 178


Adding absolute functions 179

Chapter Review

180

5. Probability and simulation

Home Page:

The heart of the matter 185

Prep Zone

186

Replay File

1865.1 Simple experiments 1875.2 Simulations 1905.3 Probability of simple events 1965.4 Sets and Venn diagrams 2005.5 Probability tables and tree diagrams 2065.6 The addition rule 2145.7 Conditional probability and the multiplication

rule 2165.8 Independent events 221

SAC Analysis Task:

Application questions 226


Making triangles 227

Chapter Review

228

MM1&2_Prelims_3pp Page iii Friday, August 12, 2005 4:30 PM

Heinemann VCE Z

ONE

: M

ATHEMATICAL

M

ETHODS

1 & 2

iv

6. Rates of change

Home Page:

How fast is too fast? 233

Prep Zone

234

Replay File

2346.1 Rates of change 2356.2 Rates of change of

linear functions 2386.3 Variable rates of change 2426.4 Displacement–time graphs 2506.5 Velocity–time graphs 254

SAC Analysis Task:

Item response 257


The great race 258

Chapter Review

260

7. Calculus

Home Page:

Where am I? 267

Prep Zone

268

Replay File

2687.1 Continuity and

hybrid functions 2697.2 Limits and theorems 2747.3 The gradient function 2817.4 Differentiation of polynomial functions

by rule 2897.5 More differentiation 2957.6 Conditions for differentiability 2987.7 Equations of tangents and normals 3007.8 Applications of calculus to curve sketching 3057.9 Types of stationary points 3097.10 Maximum/minimum problems 3157.11 Rates of change 3217.12 Antidifferentiation 323

SAC Analysis Task:

Investigation 329


Approaching infinity 330

Chapter Review

331

8. Circular functions

Home Page:

Listening to trigonometry? 337

Prep Zone

338

Replay File

3388.1 Trigonometric ratios 3398.2 Radians 3438.3 Exact values 3468.4 Unit circle: finding angles in other

quadrants 3488.5 Graphs of the sine function 3538.6 Graphs of the cosine function 3578.7 Graphs of the tangent function 3608.8 Identities 3628.9 Trigonometric equations 3648.10 Solving trigonometric problems 367

SAC Analysis Task:

Assignment 371


Sine and cosine graphs 372

Chapter Review

373

9. Exponential and logarithmic functions

Home Page:

Car stereos that would blow your mind! 377

Prep Zone

378

Replay File

3789.1 Index laws 3799.2 Rational exponents 3869.3 Solving indicial equations 3909.4 Graphing exponential functions 3939.5 Logarithms 3999.6 Logarithmic laws 4009.7 Solving logarithmic equations 4049.8 Solving indicial equations and inequalities

using logarithms 4079.9 Graphing logarithmic functions 4119.10 Exponential models 415

SAC Analysis Task:

Application questions 423


Violent logarithms 425

Chapter Review

426

10. Probability applications

Home Page:

The code breakers 431

Prep Zone

432

Replay File

43210.1 Counting

techniques 43310.2 Counting with restrictions 44110.3 Combinations and the

n

C

r

formula 44910.4 Pascal’s triangle 45510.5 Probability applications of counting

techniques 45710.6 Applications of conditional probability 461

SAC Analysis Task:

Item response 465


Winning the championship 466

Chapter Review

467

Study Guide

1. Linear functions

Summary 472Frequently Asked Questions 472Study Notes 473Cumulative Practice Examination 1 473Cumulative Practice Examination 2 473

MM1&2_Prelims_3pp Page iv Friday, August 12, 2005 4:30 PM

v

2. Quadratic functions


3. Cubic functions


4. Advanced functions and relations


5. Probability and simulation


6. Rates of change


7. Calculus


8. Circular functions


9. Exponential and logarithmic functions


10. Probability applications


Answers

517

Glossary and index

571

Notes

575

Tear-out order form for student products

581

MM1&2_Prelims_3pp Page v Friday, August 12, 2005 4:30 PM

148

Heinemann VCE Z

ONE

: M

ATHEMATICAL

M

ETHODS

1 & 2

Prepare for this chapter by attempting the following questions. If you have difficulty with a question, click on the Replay Worksheet icon on your Student CD or ask your teacher for the Replay Worksheet. Fully worked solutions to

every

question in this Prep Zone are contained in the Student

Worked Solutions book. See the order form at the back of this textbook or go to

www.hi.com.au/vcezonemaths

for further details.

1

Rearrange the following equations to make

y

the subject.

(a)

2

x

=

3

y

−

5

(b)

y

2

=

x

+

3

(c)

y

2

+

4

=

2

x

2

2

Find the equation of the straight line fitting the following criteria.

(a)

m

=

3 and

y

-intercept of 2

(b)

passing through (1, 2) and (-3, 5)

(c)

m

=

2 and passing through (-1, 4)

(d)

passing through (0, -4) and (3, 2)

3

Find the equation of each of the following quadratics.

(a)

turning point of (-1, 3) and passing through (0, 0)

(b)

passing through (1, 3), (-2, 5) and (0, 6)

4

Find the values of

x

for which each of the following is undefined.

(a) (b)

(c) (d)

5

Solve each of the following quadratic equations, leaving your answer in exact form where appropriate.

(a)

(

x

−

1)

2

=

0

(b)

(

x

+

1)

2

=

4

(c)

x

2

+

3

x

−

4

=

0

(d)

2

x

2

+

3

x

−

4

=

0

Worksheet R4.1e

Worksheet R4.2e

Worksheet R4.3e

Worksheet R4.4e2x--- x

2x 3+----------- x 4–

Worksheet R4.5e

eTestere eTestere e Interactive

Standard form of a linear equation is y = mx + c.Turning point form of a quadratic equation is y = a(x − b)2 + c , with a turning point at (b, c).

To change a quadratic to turning point form, complete the square.A fraction is not defined if its denominator equals zero.The square root of a negative number has no real number solutions.Null factor law:

If a × b = 0 then a = 0, b = 0 or a and b = 0.

y

x

y = a(x − b)2 + c

(b, c)

MM1&2_chapter04_3pp.fm Page 148 Friday, August 12, 2005 4:03 PM

4

●

advanced functions and relations

149

A

quartic

is a polynomial that has a degree of 4. The general equation for a quartic is

y

=

ax

4

+

bx

3

+

cx

2

+

dx

+

e

.

As you have seen, there is variety in the forms of cubics, and the same is true of quartics. The following shows you some of the different shapes that quartic graphs take.

y

=

x

4

y

=

x

4

+

4

x

3

−

2

x

2

−

12

x

+

9

y

=

x

4

−

8

x

3

+

18

x

2

−

27

y

=

-

x

4

−

x

3

+

7

x

2

+

x

−

6

y = -x4 + 4x3 − 3x2 − 4x + 4

As you can see, the shapes of the graphs vary considerably. Important points to notice on each of the graphs are the number of intercepts, number of turning points, and any points of inflection, including where these occur.

It is much easier to see some of these features when the functions are in factorised form. For example:

y = x4. This is the basic quartic graph. It has one turning point, at the origin.

y = x4 + 4x3 − 2x2 − 12x + 9 can be factorised to y = (x − 1)2(x + 3)2. It has two x-intercepts atx = 1 and x = -3 that are also turning points.

Generalising this, a turning point will occur at x = a when y = (x − a)2(x + b)(x + c) and in the above case both (x − 1) and (x + 3) were repeated factors.

y = x4 − 8x3 + 18x2 − 27 = (x − 3)3(x + 1). From the graph it is possible to see that there is a point of inflection when x = 3. This is because of the effect of the cubic repeated factor.

Try a few other graphs on your calculator with cubed factors as part of the quartic function.

y = -x4 − x3 + 7x2 + x − 6 = -(x + 1)(x − 2)(x + 3)(x − 1)Significantly, the intercepts occur at x = -1, 2, -3 and 1, and the graph is inverted as compared to the previous graphs. The graph is inverted because of the negative coefficient of the x4 term; the intercepts occur at each of the points where a factor equals zero.

y = -x4 + 4x3 − 3x2 − 4x + 4 = -(x − 2)2(x + 1)(x − 1)This graph has a square repeated factor and hence a turning point and intercept at x = 2. It is a negative quartic, so the graph is inverted, and the other intercepts are at x = -1 and x = 1.

Quartics4.1

-4 2 6-2

2

-24-6

y

x0

-4 2 6-2

10

4-6 0

y

x

5

15

-4 2 6-2

10

-10

-20

4-6

y

x0

-4 2 6-2

10

-10

-20

4-6

y

x0 -4 2 6-2

10

-10

-20

4-6

y

x0

A square repeated factor means that a turning point will occur at the x-intercept. The x-intercept is found by making the repeated factor equal to zero.


150 Heinemann VCE ZONE: MATHEMATICAL METHODS 1 & 2

Sketch the graph of each of the following.(a) y = x4 + 5 (b) y = (x + 2)3(x − 4)(c) y = (x − 3)2(x + 2)(x + 1) (d) y = -(x +3)2(2x + 1)2

Steps Solutions(a) 1. Determine the features of the graph. (a) y = x4 + 5

Graph of x4 translated up 5 2. Sketch the graph.

(b) 1. Determine the features of the graph. (b) y = (x + 2)3(x − 4)Graph has a point of inflection at the intercept x = -2Other x-intercept is at (4, 0)y-intercept (x = 0): 23 × -4 = -32∴ (0, -32)

2. Sketch the graph.

(c) 1. Determine the features of the graph. (c) y = (x − 3)2(x + 2)(x + 1)Repeated factor at x = 3, therefore turning point and x-intercept at (3, 0)Other x-intercepts (-2, 0) and (-1, 0)y-intercept (x = 0): (-3)2 × 2 × 1 = 18∴ (0, 18)


(d) 1. Determine the features of the graph. (d) y = -(x + 3)2(2x + 1)2

There are two repeated factors sox-intercepts and turning points at x = -3 and x = -A negative quartic so the graph is invertedy-intercept (x = 0): -32 × 12 = -9∴ (0, -9)

1

5

yy

x0

-2-32

4

y

x0

-2 3-1

y

x0

18

12--

worked example 1


4 ● advanced functions and relations 151

It is possible to obtain these graphs quickly and easily using the graphics calculator but it is important for you to recognise the features of the graphs shown in the worked example from the equations, especially when they are in factorised form.


-

-9

-3

y

x012

Use your graphics calculator to help draw the graph of each of the following, showing all the main features.(a) y = -x4 + x3 + 2x + 1 (b) y = x4 − 3x2 + 2x − 2

Steps Solutions(a) 1. Enter the equation into on your

calculator.(a)

2. Find the intercepts using

(CALC) (zero) and entering values to the left and right of where the intercept

occurs and pressing for the ‘Guess?’.Repeat for the other intercept. x-intercepts at (-0.43, 0) and (1.79, 0)

3. Find the coordinates of the maximum turning point using:

(CALC) (maximum) and again entering values to the left and

right and pressing for the ‘Guess?’. Maximum turning point at (1.14, 3.07)

4. Determine the y-intercept from the equation.

y-intercept at (0, 1)

5. Draw the graph showing these main features.

Y=

2nd TRACE

2

ENTER

2nd TRACE 4

ENTER

-0.43 1.79

y

x01

(1.14, 3.07)

worked example 2


152

Heinemann VCE Z

ONE

: M

ATHEMATICAL

M

ETHODS

1 & 2

Short answer

1

Sketch the graph of each of the following.

(a)

y

=

x

4

−

3

(b)

y

=

(

x

+

2)(

x

−

1)

3

(c)

y

=

(

x

+

3)

2

(

x

−

4)

2

(d)

y

=

(

x

+

3)(

x

−

1)

2

(

x

−

4)

(e)

y

=

(

x

+

3)(2

x

−

3)(

x

+

1)(

x

−

4)

(f)

y

=

(3

−

x

)(3

x

+

5)(

x

−

5)

2

2

Use your graphics calculator to help draw the graph of each of the following, showing all the main features.

(a)

y

=

x

4

+

3

x

2

−

2

x

+

3

(b)

y

=

-2

x

4

+

3

x

3

−

2

x

2

+

x

+

5

(c)

y

=

3

x

4

+

3

x

3

−

8

x

+

2

(d)

y

=

x

4

−

2

x

2

+

5

(e)

y

=

x

4

+

3

x

3

−

7

x

+

2

(f)

y

=

-

x

4

−

3

x

3

+

2

x

2

+

x

−

6

3

(a)

Find the equation of a quartic, with

x

4

coefficient of 1, that has an

x

-intercept and turning point at (

a

, 0) and the other two

x

-intercepts at (

p

, 0) and (

q,

0).

(b)

Expand the equation found in part

(a)

and hence find the

y

-intercept.

(b) 1. Enter the equation into on your calculator.

(b)

2. Determine the values of the x-intercepts. x-intercepts at (-2.10, 0) and (1.59, 0)3. Find the coordinates of the two minimum

turning points and the maximum turning point.

minimum turning points at (-1.37, -6.85) and (1, -2)maximum turning point at (0.37, -1.65)

4. Determine the y-intercept from the equation.

y-intercept at (0, -2)

5. Draw the graph showing these main features.

Y=

-2.1-2

1.59

(0.37, -1.65)

(1, -2)

(-1.37, -6.85)

y

x0

e Casio

e Casio

It is possible to miss a turning point on a calculator. If you are uncertain, make sure that you change the window settings appropriately.

exercise 4.1 QuarticsInteractivee

Worked Example 1

Hinte

Worked Example 2

Worked Solution

MM1&2_chapter04_3pp.fm Page 152 Wednesday, August 24, 2005 6:00 PM


Answer each of the following questions as thoroughly as possible, showing all working.1 (a) A function passes through the points (0, -5), (1, 0), (2, 7). Use finite differences to

determine the equation of the function.(b) Draw the graph of the function.(c) Draw the inverse on the graph.(d) Show at least two different restrictions to the domain that could be made so that the

inverse is a function.(e) Find the equation of the inverse.(f) Show using the domains found in part (d) that the domain of the function is the

range of the inverse and that the range of the function is the domain of the inverse.

2 Consider the graph f(x) = .

(a) Find the equation of the new function g(x) that is f(x) dilated by a factor of 2 from the x-axis, translated 3 units to the left and translated 4 units up.

(b) Use three coordinate points of f(x) and the equation of g(x) to show that g(x) = 2f(x + 3) + 4.

(c) Sketch the graph of f(x) and g(x) on the same set of axes.3 (a) Determine the equation of the graph

shown.(b) Find the equation of this graph if it is

reflected in the y-axis.(c) Find the equation of this graph if it is

reflected in the x-axis.

4 (a) Sketch the graph f(x) = 3x − 4.(b) Determine the inverse function, f-1(x).(c) Find |f-1(x)| and add this to the graph.

5 Find the equation for each of the following.(a) a rectangular hyperbola with asymptotes x = 3 and y = 2 passing through (2, -2)(b) the inverse function of f(x) = x2 + 5x + 6(c) a quartic function with turning point at (3, 0) and intercepts at (2, 0) and (-1, 0) that

passes through (4, 20).

1x2----

1 32 4 5 6 7 8 x

2

1

3

4

56

7

8

y

0-7

-7

-8

-8

-5

-5

-6

-6

-4

-4

-3

-3

-2

-2-2.5

-1-1


4 ● advanced functions and relations 179

Adding absolute functionsIf we have defined Y1 = x and Y2 = x + 5 we can then define Y3 = Y1 + Y2. We can use our graphics calculator to check that Y3 is in fact equal to 2x + 5.

1 Do this now by entering Y1 to Y3 as given and then entering Y4 as 2x + 5. How can you tell from the graph display that Y3 = 2x + 5?

If we do the same thing with absolute value functions will we get the same result?2 Set Y1 = abs(x), Y2 = abs(x + 5), Y3 = Y1 + Y2, and Y4 = abs(2x + 5) and draw

the graphs. Is Y1 + Y2 = Y4? If you have done this correctly you should have a screen display like the one shown at right. Since there are four clearly different graphs on display this means that ≠

3 To help see what is happening here, draw up a table of values for each of the equations that covers -10 � x � 10. The outline of the table is shown below.

4 The graph of consists of three sections. At which x values do the changes in sections occur? What is significant about these two values?

5 Sketch the following graphs and see if you can predict the location of the section changes in each case. You may wish to use your graphics calculator to help you with this. Make a general statement about what you find.(a) y = (b) y = (c) y = (d) y = (e) y = (f) y = (g) y = (h) y = (i) y = (j) y = (k) y = (l) y =

6 (a) Is = ? (b) Is = ?(c) What can you conclude from this?

7 Now look back at the graphs in Question 5 and see if you can work out the gradient of each of the different pieces of the graphs.

What will happen if we extend this to the addition of more than two absolute value expressions?8 Sketch each of the following graphs.

(a) y = (b) y = (c) y = (d) y =

9 What general conclusions can you draw from all of this?

x Y1 = |x| Y2 = |x + 5| Y3 = Y1 + Y2 Y4 = |2x + 5|

-10

-9

...

9

10

x x 5++ 2x 5+ .

x x 5++

x x 3++ x x 1++ x x 3–+ x x 1–+2x x 3++ 3x x 1++ 4x x 3–+ 5x x 1–+x 1+ x 3++ x 2– x 1++ x 1– x 3–+ x 2+ x 1–+

x x 3++ x 3+ x+ 2x x 3++ x 3+ 2x+

x x 1+ x 2++ + 2x 3x 1– 2x 5++ +x x 3+ x 1– x 4–+ + + 2x x 1– 3x 6– 4x 3–+ + +



See the Study Guide section for this chapter at the end of this textbook for a Chapter Summary (p. 482), Frequently Asked Questions (p. 483), Study Notes (p. 483) and Cumulative Practice Examinations (p. 484).

Use the following to check your progress. If you need more help with any questions, turn back to the section given in the side column, look carefully at the explanation of the skill and the worked examples, and try a few similar questions from the exercise provided. Fully worked solutions to every question in this Chapter Review are contained in the Student Worked Solutions book. See the order form at the back of this textbook or go to www.hi.com.au/vcezonemaths for further details.

Short answer 1 Sketch the graph of each of the following.

(a) y = x4 + 3 (b) y = x(x − 1)2(x + 3) (c) y = (x + 3)4 − 22 Find the solution(s) to each of the following polynomial equations.

(a) x4 − 3x3 + 2x2 = 0 (b) x4 + 3x2 − 4 = 0 (c) x4 + 5x3 + 5x2 − 5x − 6 = 03 Sketch the graphs for the following hyperbolas, and state the equations of the asymptotes.

(a) y = (b) y = (c) y = - (d) y = -

4 Sketch the graph for each of the following.

(a) y = + 2 (b) y = − 4 (c) y = + 2

5 State the transformations required to form the following functions from f(x) = .

(a) f(x) = + 4 (b) f(x) = − 2 (c) f(x) = + 2

6 State the equation of each of the following circles.(a) radius 2 with centre (4, 5) (b) radius 2.5 with centre (-2, 1)

7 Sketch the graph for each of the following circles, stating the radius and the centre.(a) x2 + y2 − 6x − 4y + 7 = 0 (b) x2 + y2 + 8x − 2y + 15 = 0

8 Find the inverse function for each of the following, stating the domain and range of the function f -1(x).

(a) f(x) = (b) f(x) = 4x + 5 (c) f(x) =

9 Sketch the graphs of the following absolute value functions.(a) y = (b) y = (c) y =

10 Sketch the graphs of the following quadratic absolute value functions.(a) y = (b) y = (c) y =

Multiple choice11 The equation of the graph shown could be:

A y = (x − a)2(x − b)(x + c)B y = (x − a)3(x − b)C y = (x − a)3(x + b)D y = (x + a)4

E y = (x + a)2(x + b)2

4.1

4.2

4.3

1x 3–----------- 1

3x 3+--------------- 1

x 5+----------- 1

3x 4+---------------

4.3

1x 4–( )2

------------------ 2x 4+( )2

------------------- -13 x 2–( )2----------------------

4.4x

3 x 1– 4x 8+ x 3+

4.5

4.5

4.6

3x 4+----------- x 5+

4.7x 4+ 3x 6– - x 4+ 2–

4.7x2 3+ x 3+( )2 2– 2+ - x 2+( )2 1–

ab

y

x

4.1


Documents

1. Linear functions Home Page: y a x h k 122 SAC Analysis ... · PDF file1. Linear functions Home Page: ... 1.6 Linear modelling 25 ... This is the basic quartic graph. It has one