Jacaranda Maths Quest Units 1&2 Mathematical Methods 11 for Queensland

Title page
Copyright page
Contents
About this resource
About eBookPLUS and studyON
Acknowledgements
CHAPTER 1 Arithmetic sequences
	1.1 Overview
		1.1.1 Introduction
	1.2 Arithmetic sequences
		1.2.1 Defining mathematical sequences
		1.2.2 Sequences expressed as functions
		1.2.3 Arithmetic sequences
		1.2.4 The recursive definition of arithmetic sequences
	1.3 The general form of an arithmetic sequence
		1.3.1 The general term of an arithmetic sequence
		1.3.2 Graphical display of sequences
	1.4 The sum of an arithmetic sequence
		1.4.1 Arithmetic sequences and series
		1.4.2 A visual explanation of Sn
	1.5 Applications of arithmetic sequences
		1.5.1 Simple interest
		1.5.2 Depreciating assets
	1.6 Review: exam practice
	Answers
REVISION UNIT 1 Algebra, statistics and functions
	TOPIC 1 Arithmetic and geometric sequences and series 1
CHAPTER 2 Functions
	2.1 Overview
		2.1.1 Introduction
	2.2 Functions and relations
		2.2.1 Set and interval notation
		2.2.2 Relations
		2.2.3 Functions
	2.3 Function notation
		2.3.1 Domain and range
		2.3.2 Function notation
		2.3.3 Formal mapping notation
	2.4 Transformations of functions
		2.4.1 Dilations
		2.4.2 Dilation from the x-axis by factor a
		2.4.3 Dilation from the y-axis by factor b
		2.4.4 Reflections
		2.4.5 Translations
		2.4.6 Combinations of transformations
	2.5 Piece-wise functions
		2.5.1 Piece-wise functions
		2.5.2 Modelling with piece-wise functions
	2.6 Review: exam practice
	Answers
CHAPTER 3 Quadratic relationships
	3.1 Overview
		3.1.1 Introduction
	3.2 Graphs of quadratic functions
		3.2.1 The graph of y = x2 and transformations
		3.2.2 Sketching parabolas from their equations
		3.2.3 The general, or polynomial form, y = ax2 + bx + c
		3.2.4 Turning point form, y = a(x − b)2 + c
		3.2.5 Factorised, or x-intercept, form y = a(x − b)(x − c)
		3.2.6 Determining the rule of a quadratic polynomial from a graph
		3.2.7 Using simultaneous equations
	3.3 Solving quadratic equations with rational roots
		3.3.1 Quadratic equations and the Null Factor Law
		3.3.2 Using the perfect square form of a quadratic
		3.3.3 Equations that reduce to quadratic form
	3.4 Factorising and solving quadratics over R
		3.4.1 Factorisation over R
		3.4.2 The quadratic formula
	3.5 The discriminant
		3.5.1 Defining the discriminant
		3.5.2 The role of the discriminant in quadratic equations
		3.5.3 The discriminant and the x-intercepts
		3.5.4 Intersections of lines and parabolas
	3.6 Modelling with quadratic functions
		3.6.1 Quadratically related variables
		3.6.2 Maximum and minimum values
	3.7 Review: exam practice
	Answers
CHAPTER 4 Inverse proportions and graphs of relations
	4.1 Overview
		4.1.1 Introduction
	4.2 The hyperbola
		4.2.1 The graph of y =1x
		4.2.2 General equation of a hyperbola
		4.2.3 Finding the equation of a hyperbola
		4.2.4 Modelling with the hyperbola
	4.3 Inverse proportion
	4.4 The circle
		4.4.1 Equation of a circle
		4.4.2 Semicircles
	4.5 The sideways parabola
		4.5.1 The relation y2 = x
		4.5.2 Transformations of the graph of y2 = x
		4.5.3 Determining the rule for the sideways parabola
	4.6 Review: exam practice
	Answers
CHAPTER 5 Powers and polynomials
	5.1 Overview
		5.1.1 Introduction
	5.2 Polynomials
		5.2.1 Classification of polynomials
		5.2.2 Polynomial notation
		5.2.3 Identity of polynomials
		5.2.4 Expansion of cubic and quadratic polynomials from factors
		5.2.5 Operations on polynomials
		5.2.6 Division of polynomials
	5.3 Graphs of cubic polynomials
		5.3.1 The graph of y = x3 and transformations
		5.3.2 Cubic graphs with three x-intercepts
		5.3.3 Cubic graphs with two x-intercepts
		5.3.4 Cubic graphs in the general form y = ax3 + bx2 + cx + d
		5.3.5 Determining the equation of cubic graph
	5.4 The factor and remainder theorems
		5.4.1 The remainder theorem
		5.4.2 The factor theorem
		5.4.3 Factorising polynomials
	5.5 Solving cubic equations
		5.5.1 Polynomial equations
		5.5.2 Solving cubic equations using the Null Factor Law
		5.5.3 Intersections of cubic graphs with linear and quadratic graphs
	5.6 Cubic models and applications
	5.7 Graphs of quartic polynomials
		5.7.1 Graphs of quartic polynomials of the form y = a(x − b)4 + c
		5.7.2 Quartic polynomials which can be expressed as the product of linear factors
	5.8 Solving polynomial equations
		5.8.1 The method of bisection
		5.8.2 Using the intersections of two graphs to estimate solutions to equations
		5.8.3 Estimating coordinates of turning points
	5.9 Review: exam practice
	Answers
REVISION UNIT 1 Algebra, statistics and functions
	TOPIC 2 Functions and graphs
PRACTICE ASSESSMENT 1
	Mathematical Methods: Problem solving and modelling task
CHAPTER 6 Counting and probability
	6.1 Overview
		6.1.1 Introduction
	6.2 Fundamentals of probability
		6.2.1 Notation and fundamentals: outcomes, sample spaces and events
		6.2.2 Venn diagrams
		6.2.3 Probability tables
		6.2.4 Tree diagrams
	6.3 Relative frequency
		6.3.1 Relative frequency
	6.4 Conditional probability
		6.4.1 Introduction
		6.4.2 Formula for conditional probability
		6.4.3 Multiplication of probabilities
		6.4.4 Probability tree diagrams
	6.5 Independence
		6.5.1 Introduction
		6.5.2 Test for mathematical independence
		6.5.3 Independent trials
	6.6 Permutations and combinations
		6.6.1 Arrangements or permutations
		6.6.2 Arrangements in a circle
		6.6.3 Arrangements with objects grouped together
		6.6.4 Arrangements where some objects may be identical
		6.6.5 Combinations or selections
	6.7 Pascal’s triangle and binomial expansions
		6.7.1 Pascal’s triangle
		6.7.2 Formula for binomial coefficients
		6.7.3 Pascal’s triangle with combinatoric coefficients
		6.7.4 Extending the binomial expansion to probability
	6.8 Review: exam practice
	Answers
REVISION UNIT 1 Algebra, statistics and functions
	TOPIC 3 Counting and probability
CHAPTER 7 Indices
	7.1 Overview
		7.1.1 Introduction
	7.2 Index laws
		7.2.1 Introduction
		7.2.2 Review of the index laws
		7.2.3 Products and quotients
	7.3 Negative and rational indices
		7.3.1 Negative indices
		7.3.2 Fractional indices
	7.4 Indicial equations and scientific notation
		7.4.1 Indicial equations
		7.4.2 Method of equating indices
		7.4.3 Indicial equations which reduce to quadratic form
		7.4.4 Scientific notation (standard form)
		7.4.5 Significant figures
	7.5 Review: exam practice
	Answers
REVISION UNIT 1 Algebra, statistics and functions
	TOPIC 4 Exponential functions 1
CHAPTER 8 Geometric sequences
	8.1 Overview
		8.1.1 Introduction
	8.2 Recursive definition and the general term of geometric sequences
		8.2.1 Common ratio of geometric sequences
		8.2.2 The recursive definition of a geometric sequence
		8.2.3 The general term of the geometric sequence
	8.3 The sum of a geometric sequence
		8.3.1 Limiting behaviour as n → ∞
		8.3.2 The sum of the first n terms of a geometric sequence
		8.3.3 The sum of an infinite geometric sequence
	8.4 Geometric sequences in context
		8.4.1 Growth and decay in the real world
		8.4.2 Compound interest
		8.4.3 Reducing balance depreciation
	8.5 Review: exam practice
	Answers
REVISION UNIT 1 Algebra, statistics and functions
	TOPIC 5 Arithmetic and geometric sequences and series 2
PRACTICE ASSESSMENT 2
	Mathematical Methods: Unit 1 examination
CHAPTER 9 Exponential and logarithmic functions
	9.1 Overview
		9.1.1 Introduction
	9.2 Exponential functions
		9.2.1 The graph of y = ax where a > 1
		9.2.2 The graph of y = ax where 0 < a < 1
		9.2.3 Translations of exponential graphs
	9.3 Logarithmic functions
		9.3.1 Defining logarithms
		9.3.2 Logarithm laws
		9.3.3 The graph of y = loga(x) for a > 1
		9.3.4 Extension: Transformations of logarithmic graphs
	9.4 Modelling with exponential functions
		9.4.1 Exponential growth and decay models
		9.4.2 Analysing data
	9.5 Solving equations with indices
		9.5.1 Logarithms as operators
		9.5.2 Equations containing logarithms
	9.6 Review: exam practice
	Answers
REVISION UNIT 2 Calculus and further functions
	TOPIC 1 Exponential functions 2
	TOPIC 2 The logarithmic function 1
CHAPTER 10 Trigonometric functions
	10.1 Overview
		10.1.1 Introduction
	10.2 Trigonometry review
		10.2.1 Right-angled triangles
		10.2.2 Exact values for trigonometric ratios of 30°, 45°, 60°
		10.2.3 Deducing one trigonometric ratio from another
		10.2.4 Area of a triangle
	10.3 Radian measure
		10.3.1 Definition of radian measure
		10.3.2 Extended angle measure
		10.3.3 Using radians in calculations
	10.4 Unit circle definitions
		10.4.1 Trigonometric points
		10.4.2 Unit circle definitions of the sine, cosine and tangent functions
		10.4.3 Unit circle definition of the tangent function
		10.4.4 Domains and ranges of the trigonometric functions
	10.5 Exact values and symmetry properties
		10.5.1 The signs of the sine, cosine and tangent values in the four quadrants
		10.5.2 The sine, cosine and tangent values at the boundaries of the quadrants
		10.5.3 Trigonometric points symmetric to [?] where? ∈ {30°, 45°, 60°,?6,?4,?3}
		10.5.4 Symmetry properties
	10.6 Graphs of the sine, cosine and tangent functions
		10.6.1 The graphs of y = sin(x) and y = cos(x)
		10.6.2 One cycle of the graph of y = sin(x)
		10.6.3 One cycle of the graph of y = cos(x)
		10.6.4 Guide to sketching the graphs on extended domains
		10.6.5 The graph of y = tan (x)
	10.7 Transformations of sine and cosine graphs
		10.7.1 Transformations of the sine and cosine graphs
		10.7.2 Amplitude changes
		10.7.3 Period changes
		10.7.4 Equilibrium (or mean) position changes
		10.7.5 Phase changes
		10.7.6 The graphs of y = A sin(B(x + C)) + D and y = A cos(B(x + C)) + D
		10.7.7 Forming the equation of a sine or cosine graph
	10.8 Solving rigonometric equations
		10.8.1 Solving trigonometric equations on finite domains
		10.8.2 Symmetric forms
		10.8.3 Trigonometric equations with boundary value solutions
		10.8.4 Further types of trigonometric equations
		10.8.5 Solving trigonometric equations which require a change of domain
	10.9 Modelling with trigonometric functions
		10.9.1 Maximum and minimum values
	10.10 Review: exam practice
	Answers
REVISION UNIT 2 Calculus and further functions
	TOPIC 3 Trigonometric functions 1
CHAPTER 11 Rates of change
	11.1 Overview
		11.1.1 Introduction
	11.2 Exploring rates of change
		11.2.1 Constant and variable rates of change
		11.2.2 Average rates of change
		11.2.3 Instantaneous rates of change
	11.3 The difference quotient
		11.3.1 The limit
		11.3.2 The gradient as a limit
	11.4 Differentiating simple functions
	11.5 Interpreting the derivative
		11.5.1 Interpreting the derivative as the instantaneous rate of change
		11.5.2 Interpreting the derivative as the gradient of a tangent line
	11.6 Review: exam practice
	Answers
CHAPTER 12 Properties and applications of derivatives
	12.1 Overview
	12.2 Differentiation by formula
	12.3 The derivative as a function
		12.3.1 Derivative notation
		12.3.2 Differentiability of a function
	12.4 Properties of the derivative
	12.5 Differentiation of power and polynomial functions
	12.6 Review: exam practice
	Answers
CHAPTER 13 Applications of derivatives
	13.1 Overview
		13.1.1 Introduction
	13.2 Gradient and equation of a tangent
		13.2.1 Tangents
		13.2.2 Normals
	13.3 Displacement–time graphs
		13.3.1 Definitions
	13.4 Sketching curves using derivatives
		13.4.1 Sketching curves with the first derivative test
		13.4.2 Sketching curves with the second derivative test
		13.4.3 Global maxima and minima
		13.4.4 End behaviour of a function
	13.5 Modelling optimisation problems
		13.5.1 When the rule for the function is known
		13.5.2 When the rule for the function is not known
	13.6 Review: exam practice
	Answers
REVISION UNIT 2 Calculus and further functions
	TOPIC 4 Introduction to differential calculus
CHAPTER 14 Differentiation rules
	14.1 Overview
	14.2 The product rule
	14.3 The quotient rule
	14.4 The chain rule
	14.5 Applications of the product, quotient and chain rules
	14.6 Review: exam practice
	Answers
REVISION UNIT 2 Calculus and further functions
RevisionUnit2Topic05
	TOPIC 5 Further differentiation and applications 1
CHAPTER 15 Discrete random variables 1
	15.1 Overview
		15.1.1 Introduction
	15.2 Discrete random variables
	15.3 Expected values
	15.4 Variance and standard deviation
		15.4.1 Introduction
		15.4.2 Properties of the variance
	15.5 Applications of discrete random variables
	15.6 Review: exam practice
	Answers
REVISION UNIT 2 Calculus and further functions
	TOPIC 6 Discrete random variables 1
PRACTICE ASSESSMENT 3
	Mathematical Methods: Unit 2 examination
PRACTICE ASSESSMENT 4
	Mathematical Methods: Units 1 & 2 examination
GLOSSARY
INDEX