Mathematics Methods

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand real-world phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.

Mathematics Methods Level 4 provides the study of algebra, functions, differential and integral calculus, probability and statistics. These are necessary prerequisites for the study of Mathematics Specialised Level 4 and as a foundation for tertiary studies in disciplines in which mathematics and statistics have important roles, including engineering, the sciences, commerce and economics, health and social sciences.

Mathematics Methods Level 4 aims to develop learners’:

• understanding of concepts and techniques and problem solving ability in the areas of algebra, function study, differential and integral calculus, probability and statistics
• reasoning skills in mathematical contexts and in interpreting mathematical information
• capacity to communicate in a concise and systematic manner using mathematical language.

On successful completion of this course, learners will be able to:

1. understand the concepts and techniques in algebra, graphs, function study, differential and integral calculus, probability and statistics
2. solve problems using algebra, graphs, function study, differential and integral calculus, probability and statistics
3. apply reasoning skills in the context of algebra, graphs, function study, differential and integral calculus, probability and statistics
4. interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems
5. communicate their arguments and strategies when solving problems
6. plan activities and monitor and evaluate their progress
7. use strategies to organise and complete activities to organise and complete activities and meet deadlines in the context of mathematics
8. select and use appropriate tools, including computer technology, when solving mathematical problems
9. Additionally, learners will be given opportunities to demonstrate the following in line with the Australian Curriculum General Capabilities: literacy skills; numeracy skills; information and communication technology skills; critical and creative thinking skills; ethical and intercultural understanding.

It is highly recommended that learners attempting this course will have successfully completed either of the courses Mathematics Methods – Foundation Level 3 or the AC Mathematics 10A subject with some additional studies in introductory calculus.

Assumed knowledge and skills are contained in both of these courses and will be drawn upon in the development of key knowledge and skills in Mathematics Methods.

Learners must have access to calculator algebraic system (CAS) graphics calculators and become proficient in their use. These calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems. Refer to 'What can I take to my exam?' for the current TASC Calculator Policy that applies to Level 3 and 4 courses.

The use of computer software is also recommended as an aid to students’ learning and mathematical development. A range of packages such as, but not limited to: Wolfram Mathematica, Microsoft Excel, Autograph, Efofex Stat, Graph and Draw are appropriate for this purpose.

This course has a complexity level of 4.

In general, courses at this level provide theoretical and practical knowledge and skills for specialised and/or skilled work and/or further learning, requiring:

• broad factual, technical and some theoretical knowledge of a specific area or a broad field of work and learning
• a broad range of cognitive, technical and communication skills to select and apply a range of methods, tools, materials and information to:
  • complete routine and non-routine activities
  • provide and transmit solutions to a variety of predictable and sometimes unpredictable problems
• application of knowledge and skills to demonstrate autonomy, judgement and limited responsibility in known or changing contexts and within established parameters.

This level 4 course has a size value of 15.

This course is made up of five (5) areas of study. While each of these is compulsory, the order of delivery is not prescribed.

• Function Study
• Circular (Trigonometric) Functions
• Differential Calculus
• Integral Calculus
• Probability and Statistics.

These areas of study relate to Assessment Criteria 4–8. Assessment Criteria 1–3 apply to all five areas of study.

Whilst the areas of study may be addressed separately and in any order, much of the content is inter-related and a more integrated approach is recommended and encouraged. It is also recommended that, where possible, concepts be developed within a context of practical applications. Such an approach provides learners with mathematical experiences that are much richer than a collection of skills. Learners thereby have the opportunity to observe and make connections between related aspects of the course and the real world and to develop further some important abstract ideas.

FUNCTION STUDY

Learners will develop their understanding of the behaviour of a number of different functions.

This area of study will include:

• understanding the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another
• using function notation, domain and range
• understanding the concept of the graph of a function
• reviewing the binomial theorem for `(ax + b)^n` and its link to Pascal’s triangle (formal derivation and extended problems involving particular terms not included)
• graphing polynomial functions in factored form with linear factors, including repeated factors
• reviewing the factorisation of polynomials and their use in curve sketching and determining the nature of stationary points
• recognising features and drawing graphs of `y = x^n` for `n = -2, -1, 1/2, 1, 2` and `3`, including shape and behaviour as `x -> oo` and `x -> -oo` 
• defining logarithms as indices: `a^x = b` is equivalent to `x = log_a b`, i.e. `a^(log_a b) = b` 
• establishing and using the algebraic properties of logarithms
• recognising the inverse relationship between logarithms and exponentials: `y = a^x` is equivalent to `x = log_a y` 
• solving equations involving indices using logarithms
• recognising the qualitative features of the graphs of `y = a^x` and `y = log_a x`  (`a = 2, e` and `10`) including domain, range and, where relevant, asymptotes
• solving simple equations involving simple exponential and logarithmic functions algebraically and graphically
• defining the natural logarithm `ln x = log_e x` 
• recognising and using the inverse relationship of the functions `y = e^x` and `y = ln x` 
• understanding transformations (reflections, dilations and translations) from `y = f(x)` to `y = Af[n(x + h)]+k`, where `A, n, h` and `k in R`, and `A, n != 0` 
• using the functions articulated above to model practical situations (this does not include regression analysis)
• using the composition of functions where  `f` `composition` `g`  is defined by `f (g (x))`, given `ran g sube dom f` (the notation `f@g` may be used, but is not required)
• solving equations of the form `f (x) = g (x)` over a specified interval, where `f` and `g` are functions of the styles specified above (using algebraic and calculator techniques where appropriate)
• sketching and interpreting polynomial, power, exponential and logarithmic functions, including simple transformations and combinations of these functions, including simple piecewise (hybrid) functions
• recognising the distinction between a function and a relation, including an informal consideration of one-to-one and many-to-one functions and the conditions for existence of an inverse function
• finding inverses of functions, including conditions for the existence of an inverse function, including exponential, logarithmic and power functions such as:
  `y= a(x+b)^2`       `y =a/(x+ b)+ c`        `y = a e^(x+b)+c`           `y = aln (x+b) +c` 
• finding the graphs of inverses derived from graphs of original functions.

CIRCULAR (TRIGONOMETRIC) FUNCTIONS

Learners will develop their understanding of a range of circular (trigonometric) functions.

This area of study will include:

• using the angular measurements definition of radians and its relationship with degree measure
• reviewing the trigonometrical relationships involving the unit circle:
  • the definition of sine, cosine and tangent
  • the special relationships `sin^2x + cos^2x = 1`  (Pythagorean identity) and `-1 <= sin x <= 1`,  `-1 <= cos x <= 1` 
  • the identity `tan x =sin x/ cos x` 
  • the special values; e.g. `sin(0) = 0`,  `cos(pi) = -1` 
  • the symmetry properties, such as `sin (pi +- x)`, `cos (pi +- x)`, `sin (2pi +- x)`, `cos (2pi +- x)`, `tan (pi +- x)`, `tan (2pi +- x)`, `sin (pi/2 +- x)`, `cos (pi/2 +- x)`, `tan (pi/2 +- x)` 
• recognising the exact values for sine, cosine and tangent for: `pi/6," "pi/4," "pi/3," "pi/2` and integer multiples of these
• recognising the graphs of `y = sin x`, `y = cos x` and`y = tan x` on extended domains
• examining the effect of transformations of circular functions to: `y = a sin (bx + c) + d`, `y= a cos (bx + c) + d`, `y = a tan (bx + c) + d` 
• using inverse trigonometric functions to enable the solution of trigonometric equations of the form: `sin (ax + b) = c`,  `cos (ax + b) = c`,  `tan (ax + b) = c` and also of the form `sin(ax + b )= k cos (ax + b)`  giving answers specific to a given domain
• using the functions articulated in this section to model practical situations (this does not include regression analysis).

CALCULUS

Learners will study the graphical treatment of limits and differentiability of functions of a single real variable, and the differentiation and integration of these functions.

This area of study will include:

Differential Calculus:

• reviewing the differences between the average rate of change and an instantaneous rate of change
• reviewing the concept of a limit and the evaluation of a limit, e.g. `lim _(x->a) f(x) = p` 
• applying the first principles approach, `f'(x) = lim_(h->0) (f(x+h) - f(x))/h` to determine the gradient functions of `f(x) = x^2, x^3` and other simple quadratic and cubic functions
• determining the derivatives of `x^n`, for `n in Q`, `e^x`, `log_e x`, `sin x`, `cos x` and `tan x`, and linear combinations of these functions (formal derivation not examined)
• determining the derivatives of `f (x) +- g(x)`, `f(x) xx g(x)`, `f(x)/g(x)`, and `f (g(x))` where `f` and `g` are polynomial functions, exponential, circular, logarithmic or power functions and transformations or simple transformations of these functions, or are simple manipulations of these
• consideration of the differentiability of functions
• finding the slope of a tangent and the equations of the tangent and normal to a curve at a point
• determining instantaneous rates of changes
• deducing the graph of the derivative function, including its domain, from the graph of a function
• studying applications of differentiation to graphs and identifying the key features of graphs:
  • identifying intervals over which a function is constant (stationary), strictly increasing or strictly decreasing
  • identifying local maximums / minimums / stationary points of inflections over an interval (learners can choose to use any methods as appropriate; the ‘change of sign of the first derivative’ test, the graph of the derivative, or the second derivative test, to justify these)
  • distinguishing between a local maximum / minimum and an end point maximum / minimum

• studying the applications of differentiation to solving problems
• constructing and interpreting position-time graphs, with velocity as the slope of the tangent.

Integral Calculus:

• understanding anti-differentiation, recognising that `F'(x) = f(x)`implies that `int f(x)dx = F(x) + c`
• determining anti-derivatives (indefinite integrals) of polynomial functions and functions of the form `f(ax + b)` where `f` is `x^n`, for `n in Q`, `e^x`, `sin (x)`, `cos (x)`, and combinations of these
• informally considering the definite integral as a limiting value of a sum involving quantities such as area under a curve and the informal treatment of the Fundamental Theorem of Calculus, `int_a^b f(x)dx = F(b) - F(a)` 
• applying integration to problems involving a function given a boundary condition, calculating the area of a region under a curve and simple cases of areas between curves
• determining the equation of a function given its gradient or the equation of a tangent at a point on the curve
• investigating applications of integration, including displacement as the integral of velocity.

PROBABILITY AND STATISTICS

Learners will study scenarios involving discrete and continuous random variables, their representation using tables, probability functions specified by rule and defining parameters (as appropriate); the calculation and interpretation of central measures and measure and spread; and statistical inference for sample proportions.

This area of study will include:

• understanding the concept of a random variable as a real function defined on a sample space, studying examples of discrete and continuous random variables
• investigating discrete random variables:
  • specifying probability distributions for discrete random variables using graphs, tables and probability mass functions
  • calculating, interpreting and using the expected value (`E`), the mean (`mu`), variance  (`sigma^2`) of a discrete random variable with and without technology
  • recognising the property that, for many random variables, approximately 95 per cent of the distribution is within two standard deviations of the mean
  • using discrete random variables and associated probabilities to solve practical problems
• applying the binomial distribution to the number of successes in a fixed number, `n`, of Bernoulli trials with probability `p` of success:
  • examining the effect of the parameters `n` and `p` on the graph of the probability function
  • calculating probabilities `P(X=r) = ((n),(r))p^r (1 - p)^(n - r)` associated with the binomial distribution with parameters `n` and `p`; noting the mean `np` and variance `np(1 - p)` of a binomial distribution
  • using binomial distributions and associated probabilities to model data and solve practical problems
• studying the normal distribution:
  • identifying contexts such as naturally occurring variation that are suitable for modelling by normal random variables
  • recognising features of the graph of the probability density function of the normal distribution with mean `mu` and standard deviation `sigma` and the use of the standard normal distribution
  • using technology to calculate the mean `mu`, variance `sigma^2` and standard deviation `sigma` from given continuous data
  • using the 68-95-99% approximation rule, to determine the proportions of a population within 1, 2 or 3 standard deviations of the mean
  • calculating probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems
  • calculating mean and standard deviation of a normal distribution given proportion information
  • recognising features of the standard normal distribution `N (0, 1)` 
• studying statistical inference, including the definition and distribution of a sample proportions, simulations and confidence intervals:
  • understanding the distinction between a population parameter and a sample statistic, using a sample statistic to estimate the population parameter
  • using the concept of the sample proportion `hat p = X/n` as a random variable whose value varies between samples, where `X` is a binomial random variable which is associated with the number of items that have a particular characteristic and n is the sample size
  • using the approximate normality of the distribution of `hat p` for large samples and, for such a situation, the mean `mu_p`, (the population proportion) and standard deviation `sigma_p = sqrt ((p(1 - p))/n)`
  • from a large sample, determining an approximate confidence interval `(hat p - z sqrt ((hat p(1 - hat p))/n)," " hat p + z sqrt ((hat p(1 - hat p))/n))` for a population proportion where `z` is the appropriate quantile for the standard normal distribution. Whilst the 95% confidence interval is a particular example of such an interval where `z ~~ 1.96` is mainly used, some consideration should be made to other confidence intervals, including the 90% and the 99%. (The terms ‘standard error’ or ‘approximate margin of error (`M`)’ may also be used).

Criterion-based assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate end-point of study. Although assessment – as part of the learning program – is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect end-point achievement.

The standard of achievement each learner attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.

A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.

A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.

Providers offering this course must participate in quality assurance processes specified by TASC to ensure provider validity and comparability of standards across all awards. To learn more, see TASC's quality assurance processes and assessment information.

Internal assessment of all criteria will be made by the provider. Providers will report the learner’s rating for each criterion to TASC. TASC will supervise the external assessment of designated criteria which will be indicated by an asterisk (*). The ratings obtained from the external assessments will be used in addition to internal ratings from the provider to determine the final award.

The following processes will be facilitated by TASC to ensure there is:

• a match between the standards of achievement specified in the course and the skills and knowledge demonstrated by learners
• community confidence in the integrity and meaning of the qualification.

Process – TASC gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.

Additionally, the Office of TASC may select to undertake scheduled audits of this course and its work requirements (Provider standards 1, 2, 3 and 4).

The external assessment for this course will comprise:

• a three hour written examination assessing criteria: 4, 5, 6, 7 and 8.

For further information, see the current external assessment specifications and guidelines for this course available in the Supporting Documents below.

The assessment for Mathematics Methods Level 4 will be based on the degree to which the learner can:

• 1. communicate mathematical ideas and information
• 2. apply mathematical reasoning and strategy in problem solving situations
• 3. use resources and organisational strategies
• 4. understand polynomial, hyperbolic, exponential and logarithmic functions*
• 5. understand circular functions*
• 6. use differential calculus in the study of functions*
• 7. use integral calculus in the study of functions*
• 8. understand binomial and normal probability distributions and statistical inference*

* = denotes criteria that are both internally and externally assessed

Mathematics Methods Level 4 (with the award of):

EXCEPTIONAL ACHIEVEMENT

HIGH ACHIEVEMENT

COMMENDABLE ACHIEVEMENT

SATISFACTORY ACHIEVEMENT

PRELIMINARY ACHIEVEMENT

The final award will be determined by the Office of Tasmanian Assessment, Standards and Certification from 13 ratings (8 from the internal assessment, 5 from the external assessment).

The minimum requirements for an award in Mathematics Methods Level 4 are as follows:

 EXCEPTIONAL ACHIEVEMENT (EA)
 11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings and 1 ‘B’ rating from external assessment)

 HIGH ACHIEVEMENT (HA)
 5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings and 1 ‘C’ rating from external assessment)

 COMMENDABLE ACHIEVEMENT (CA)
 7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings and 2 ‘C’ ratings from external assessment)

 SATISFACTORY ACHIEVEMENT (SA)
 11 ‘C’ ratings (3 ‘C’ ratings from external assessment)

 PRELIMINARY ACHIEVEMENT (PA)
 6 ‘C’ ratings

A learner who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.

The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment.

In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.

Requests for amendments to an accredited course will be forwarded by Curriculum Services to the Office of TASC for formal consideration.

Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.

A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.

The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.

For the content areas of Mathematics Methods, the proficiency strands – Understanding; Fluency; Problem Solving; and Reasoning – build on learners’ learning in F-10 Australian Curriculum: Mathematics. Each of these proficiencies is essential, and all are mutually reinforcing. They are still very much applicable and should be inherent in the five areas of study.

MATHEMATICAL METHODS

Unit 1 – Topic 1: Functions and Graphs

Lines and linear relationships:

• examine examples of direct proportion and linearly related variables (ACMMM002)
• recognise features of the graph of `y = mx + c`, including its linear nature, its intercepts and its slope or gradient (ACMMM003)
• find the equation of a straight line given sufficient information; parallel and perpendicular lines (ACMMM004)
• solve linear equations (ACMMM005).

Review of quadratic relationships:

• examine examples of quadratically related variables (ACMMM006)
• recognise features of the graphs of `y = x^2`, `y = a(x - b)^2 + c`, and `y = a(x - b)(x - c)`, including their parabolic nature, turning points, axes of symmetry and intercepts (ACMMM007)
• solve quadratic equations using the quadratic formula and by completing the square (ACMMM008)
• find the equation of a quadratic given sufficient information (ACMMM009)
• find turning points and zeros of quadratics and understand the role of the discriminant (ACMMM010)
• recognise features of the graph of the general quadratic `y = ax^2 + bx + c` (ACMMM011).

Inverse proportion:

• examine examples of inverse proportion (ACMMM012)
• recognise features of the graphs of `y = 1/x` and `y = a/(x - b)`, including their hyperbolic shapes, and their asymptotes (ACMMM013).

Powers and polynomials:

• recognise features of the graphs of `y = x^n` for `n in N`, `n = -1` and `n = 1/2`, including shape and behaviour as `x -> oo` and `x -> -oo` (ACMMM014)
• identify the coefficients and the degree of a polynomial (ACMMM015)
• expand quadratic and cubic polynomials from factors (ACMMM016)
• recognise features of the graphs of `y = x^3`, `y = a(x - b)^3 + c` and `y = k(x - a)(x - b)(x - c)`, including shape, intercepts and behaviour as `x -> oo` and `x -> -oo` (ACMMM017)
• factorise cubic polynomials in cases where a linear factor is easily obtained (ACMMM018)
• solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained (ACMMM019).

Functions:

• understand the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another (ACMMM022)
• use function notation, domain and range, independent and dependent variables (ACMMM023)
• understand the concept of the graph of a function (ACMMM024)
• examine translations and the graphs of `y = f(x) + a` and `y = f(x + b)` (ACMMM025)
• examine dilations and the graphs of `y = cf(x)` and `y = f(kx)` (ACMMM026)
• recognise the distinction between functions and relations, and the vertical line test (ACMMM027).

Unit 1 – Topic 2: Trigonometric Functions

Cosine and sine rules:

• review sine, cosine and tangent as ratios of side lengths in right-angled triangles (ACMMM028)
• understand the unit circle definition of `cos theta`, `sin theta` and `tan theta`, and periodicity using degrees (ACMMM029).

Circular measure and radian measure:

• define and use radian measure and understand its relationship with degree measure (ACMMM032).

Trigonometric functions:

• understand the unit circle definition of `cos theta`, `sin theta` and `tan theta`, and periodicity using radians (ACMMM034)
• recognise the exact values of `sin theta`, `cos theta` and `tan theta` at integer multiples of `pi/6` and `pi/4` (ACMMM035)
• recognise the graphs of `y = sin x`, `y = cos x`, and `y = tan x` on extended domains (ACMMM036)
• examine amplitude changes and the graphs of `y = a sin x` and `y = a cos x`  (ACMMM037)
• examine period changes and the graphs of `y = sin bx`, `y = cos bx`, and `y = tan bx` (ACMMM038)
• examine phase changes and the graphs of `y = sin(x + c)`, `y = cos(x + c)`, and `y = tan bx` (ACMMM039)
• `y = tan(x + c)` and the relationships `sin(x + pi/2) = cos x` and `cos(x - pi/2) = sin x` (ACMMM040)
• identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems (ACMMM042)
• solve equations involving trigonometric functions using technology, and algebraically in simple cases (ACMMM043).

Unit 1 – Topic 3: Counting and Probability

Combinations:

• understand the notion of a combination as an unordered set of `r` objects taken from a set of `n` distinct objects (ACMMM044)
• use the notation `((n),(r))` and the formula `((n),(r)) = (n!)/(r!(n - r)!)` for the number of combinations of `r` objects taken from a set of `n` distinct objects (ACMMM045)
• expand `(x + y)^n` for small positive integers `n` (ACMMM046)
• recognise the numbers `((n),(r))` as binomial coefficients (as coefficients in the expansion of `(x + y)^n` (ACMMM047).

Unit 2 – Topic 1: Exponential Functions

Indices and the index laws:

• review indices (including fractional indices) and the index laws (ACMMM061)
• use radicals and convert to and from fractional indices (ACMMM062)
• understand and use scientific notation and significant figures (ACMMM063).

Exponential functions:

• establish and use the algebraic properties of exponential functions (ACMMM064)
• recognise the qualitative features of the graph of `y = a^x` (`a > 0`), including asymptotes, and of its translations (`y = a^x + b` and `y = a^(x + c)`) (ACMMM065)
• identify contexts suitable for modelling by exponential functions and use them to solve practical problems (ACMMM066)
• solve equations involving exponential functions using technology, and algebraically in simple cases (ACMMM067).

Unit 2 – Topic 3: Introduction to Differential Calculus

Rates of change:

• interpret the difference quotient `(f(x + h) - f(x))/h` as the average rate of change of a function `f` (ACMMM077)
• use the Leibniz notation `delta x` and `delta y` for changes or increments in the variables `x` and `y` (ACMMM078)
• use the notation `(delta y)/(delta x)` for the difference quotient `(f(x + h) - f(x))/h` where `y = f(x)` (ACMMM079)
• interpret the ratios `(f(x + h) - f(x))/h` and `(delta y)/(delta x)` as the slope or gradient of a chord or secant of the graph of `y = f(x)` (ACMMM080).

The concept of the derivative:

• examine the behaviour of the difference quotient `(f(x + h) - f(x))/h` as `h -> 0` as an informal introduction to the concept of a limit (ACMMM081)
• define the derivative `f'(x)` as `lim_(h -> 0) (f(x + h) - f(x))/h` (ACMMM082)
• use the Leibniz notation for the derivative: `dy/dx = lim_(delta x -> 0) (delta y)/(delta x)` and the correspondence `dy/dx = f'(x)` where `y = f(x)` (ACMMM083)
• interpret the derivative as the instantaneous rate of change (ACMMM084)
• interpret the derivative as the slope or gradient of a tangent line of the graph of `y = f(x)` (ACMMM085).

Computation of derivatives:

• estimate numerically the value of a derivative, for simple power functions (ACMMM086)
• examine examples of variable rates of change of non-linear functions (ACMMM087)
• establish the formula `d/dx(x^n) = nx^(n - 1)` for positive integers `n` by expanding `(x + h)^n` or by factorising `(x + h)^n - x^n` (ACMMM088).

Properties of derivatives:

• understand the concept of the derivative as a function (ACMMM089)
• recognise and use linearity properties of the derivative (ACMMM090)
• calculate derivatives of polynomials and other linear combinations of power functions (ACMMM091).

Applications of derivatives:

• find instantaneous rates of change (ACMMM092)
• find the slope of a tangent and the equation of the tangent (ACMMM093)
• construct and interpret position-time graphs, with velocity as the slope of the tangent (ACMMM094)
• sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as `x -> oo` and `x -> -oo` (ACMMM095)
• solve optimisation problems arising in a variety of contexts involving simple polynomials on finite interval domains (ACMMM096).

Anti-derivatives:

• calculate anti-derivatives of polynomial functions and apply to solving simple problems involving motion in a straight line (ACMMM097).

Unit 3 – Topic 1: Further Differentiation and Applications

Exponential functions:

• estimate the limit of `(a^h - 1)/h` as `h -> 0` using technology, for various values of `a > 0` (ACMMM098)
• recognise that `e` is the unique number `a` for which the above limit is 1 (ACMMM099)
• establish and use the formula `d/dx(e^x) = e^x` (ACMMM100)
• use exponential functions and their derivatives to solve practical problems (ACMMM101).

Trigonometric functions:

• establish the formulas `d/dx(sin x) = cos x`, and `d/dx(cos x) = -sin x` by numerical estimations of the limits and informal proofs based on geometric constructions (ACMMM102)
• use trigonometric functions and their derivatives to solve practical problems (ACMMM103).

Differentiation rules:

• understand and use the product and quotient rules (ACMMM104)
• understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions (ACMMM105)
• apply the product, quotient and chain rule to differentiate functions such as `xe^x`, `tan x`, `1/(x^n)`, `x sin x`, `e^(-x) sin x` and `f(ax + b)` (ACMMM106).

The second derivative and applications of differentiation:

• understand the concept of the second derivative as the rate of change of the first derivative function (ACMMM108)
• recognise acceleration as the second derivative of position with respect to time (ACMMM109)
• understand the concepts of concavity and points of inflection and their relationship with the second derivative (ACMMM110)
• understand and use the second derivative test for finding local maxima and minima (ACMMM111)
• sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as `x -> oo` and `x -> -oo` (ACMMM095)
• solve optimisation problems arising in a variety of contexts involving simple polynomials on finite interval domains (ACMMM096).

Unit 3 – Topic 2: Integrals

Anti-differentiation:

• recognise anti-differentiation as the reverse of differentiation (ACMMM114)
• use the notation `int f (x)dx` for anti-derivatives or indefinite integrals (ACMMM115)
• establish and use the formula `int (x^n)dx = 1/(n + 1) x^(n + 1) + c` for `n != -1` (ACMMM116)
• establish and use the formula `int e^x dx = e^x + c` (ACMMM117)
• establish and use the formulae `int sin x dx = -cos x + c` and `int cos x dx = sin x + c` (ACMMM118)
• recognise and use linearity of anti-differentiation (ACMMM119)
• determine indefinite integrals of the form `int f(ax + b)dx` (ACMMM120)
• identify families of curves with the same derivative function (ACMMM121)
• determine `f(x)`, given `f'(x)` and an initial condition `f(a) = b` (ACMMM122)
• determine displacement given velocity in linear motion problems (ACMMM123).

Definite integrals:

• examine the area problem, and use sums of the form `sum_(i) f(x_i) delta x_i` to estimate the area under the curve `y = f(x)` (ACMMM124)
• interpret the definite integral `int_a^b f(x) dx` as area under the curve `y = f(x)` if `f(x) > 0` (ACMMM125)
• recognise the definite integral`int_a^b f(x) dx` as a limit of sums of the form `sum_i f(x_i) delta x_i` (ACMMM126)
• interpret `int_a^b f(x) dx` as a sum of signed areas (ACMMM127)
• recognise and use the additivity and linearity of definite integrals (ACMMM128).

Fundamental theorem:

• understand the concept of the signed area function `F(x) = int_a^x f(t) dt`ACMMM129)
• understand and use the theorem: `F'(x) = d/dx (int_a^x f(t) dt) = f(x)`, and illustrate its proof geometrically (ACMMM130)
• understand the formula `int_a^b f(x) dx = F(b) - F(a)` and use it to calculate definite integrals (ACMMM131).

Applications of integration:

• calculate the area under a curve (ACMMM132)
• calculate the area between curves in simple cases (ACMMM134)
• determine positions given values of velocity (ACMMM135).

Unit 3 – Topic 3: Discrete Random Variables

General discrete random variables:

• understand the concepts of a discrete random variable and its associated probability function, and their use in modelling data (ACMMM136)
• use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable (ACMMM137)
• recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes (ACMMM138)
• examine simple examples of non-uniform discrete random variables (ACMMM139)
• recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases (ACMMM140)
• recognise the variance and standard deviation of a discrete random variable as a measures of spread, and evaluate them in simple cases (ACMMM141)
• use discrete random variables and associated probabilities to solve practical problems (ACMMM142).

Bernoulli distributions:

• use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)
• identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)
• recognise the mean `p` and variance `p(1 - p)` of the Bernoulli distribution with parameter `p` (ACMMM145)
• use Bernoulli random variables and associated probabilities to model data and solve practical problems (ACMMM146).

Binomial distributions:

• understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in `n` independent Bernoulli trials, with the same probability of success `p` in each trial (ACMMM147)
• identify contexts suitable for modelling by binomial random variables (ACMMM148)
• determine and use the probabilities `P(X = r) = ((n),(r)) p^r(1 - p)^(n - r)` associated with the binomial distribution with parameters `n` and `p` note the mean `np` and variance `np(1 - p)` of a binomial distribution (ACMMM149)
• use binomial distributions and associated probabilities to solve practical problems (ACMMM150).

Unit 4 – Topic 1: The Logarithmic Function

Logarithmic functions:

• define logarithms as indices: `a^x = b` is equivalent to `x = log_a b,` i.e. `a^(log_a b) = b` (ACMMM151)
• establish and use the algebraic properties of logarithms (ACMMM152)
• recognise the inverse relationship between logarithms and exponentials: `y = a^x` is equivalent to `x = log_a y` (ACMMM153)
• interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake magnitude, octaves in music, pH in chemistry (ACMMM154)
• solve equations involving indices using logarithms (ACMMM155)
• recognise the qualitative features of the graph of `y = log_a x` (`a > 1`) including asymptotes, and of its translations `y = log_a x + b` and `y = log_a (x + c)` (ACMMM156)
• solve simple equations involving logarithmic functions algebraically and graphically (ACMMM157)
• identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems (ACMMM158).

Calculus of logarithmic functions:

• define the natural logarithm `ln x = log_e x` ACMMM159)
• recognise and use the inverse relationship of the functions `y = e^x` and `y = ln x` (ACMMM160)
• establish and use the formula `d/dx (ln x ) = 1/x` (ACMMM161)
• establish and use the formula `int 1/x dx = ln x + c`, for `x > 0` (ACMMM162)
• use logarithmic functions and their derivatives to solve practical problems (ACMMM163).

Unit 4 – Topic 2: Continuous Random Variables and the Normal Distribution

General continuous random variables:

• use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable (ACMMM164)
• understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts (ACMMM165)
• understand the effects of linear changes of scale and origin on the mean and the standard deviation (ACMMM167).

Normal distributions:

• identify contexts such as naturally occurring variation that are suitable for modelling by normal random variables (ACMMM168)
• recognise features of the graph of the probability density function of the normal distribution with mean `mu` and standard deviation `sigma`, and the use of the standard normal distribution (ACMMM169)
• calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems (ACMMM170).

Unit 4 – Topic 3: Interval Estimates for Proportions

Random sampling:

• understand the concept of a random sample (ACMMM171)
• discuss sources of bias in samples, and procedures to ensure randomness (ACMMM172)
• use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli (ACMMM173).

Sample proportions:

• understand the concept of the sample proportion `hat p` as a random variable whose value varies between samples, and the formulas for the mean `p` and standard deviation `sqrt ((p(1 - p))/n)` of the sample proportion `hat p` (ACMMM174)
• examine the approximate normality of the distribution of `hat p` for large samples (ACMMM175)
• simulate repeated random sampling, for a variety of values of `p` and a range of sample sizes, to illustrate the distribution of `hat p` and the approximate standard normality of `(hat p - p)/sqrt((hat p(1 - hat p))/n)` where the closeness of the approximation depends on both `n` and `p` (ACMMM176).

Confidence intervals for proportions:

• the concept of an interval estimate for a parameter associated with a random variable (ACMMM177)
• use the approximate confidence interval `(hat p - z sqrt ((hat p(1 - hat p))/n)," " hat p + z sqrt ((hat p(1 - hat p))/n))`, as an interval estimate for `p`, where `z` is the appropriate quantile for the standard normal distribution (ACMMM178)
• define the approximate margin of error `E = z sqrt ((hat p(1 - hat p))/n)` and understand the trade-off between margin of error and level of confidence (ACMMM179)
• use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain `p` (ACMMM180).

The accreditation period for this course has been renewed from 1 January 2022 until 31 December 2025.

During the accreditation period required amendments can be considered via established processes.

Should outcomes of the Years 9-12 Review process find this course unsuitable for inclusion in the Tasmanian senior secondary curriculum, its accreditation may be cancelled. Any such cancellation would not occur during an academic year.

Version 1 – Accredited on 17 August 2016 for use from 1 January 2017. This course replaces Mathematics Methods (MTM315114) that expired on 31 December 2016.

Version 1.1 – Renewal of accreditation on 13 August 2017 for use in 2018.

Accreditation renewed on 22 November 2018 for the period 1 January 2019 until 31 December 2021.

Version 1.2 - Renewal of Accreditation on 14 July 2021 for the period 31 December 2021 until 31 December 2024, without amendments.

GLOSSARY

Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(-x) = 1/a^x`, `(a^x)^y = a^(xy)`, `a^0 = 1`, where `x`, `y` and `a` are real.

Additivity property of definite integrals
The additivity property of definite integrals refers to ‘addition of intervals of integration’: `int_a^b f(x) dx + int_b^c f(x) dx = int_a^c f(x) dx` for any numbers `a`, `b` and `c`, and any function `f(x)`.

Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(-x) = 1/a^x`, `(a^x)^y = a^(xy)`, `a^0 = 1`, for any real numbers `x`, `y`, and `a`, with `a > 0`.

Algebraic properties of logarithms
The algebraic properties of logarithms are the rules: `log_a (xy) = log_a x + log_a y`, `log_a (1/x) = -log_a x`, and `log_a 1 = 0`, for any positive real numbers `x`, `y` and `a`.

Antidifferentiation
An anti-derivative, primitive or indefinite integral of a function `f(x)` is a function `F(x)` whose derivative is `(x)`, i.e. `F'(x) = f(x)`.

The process of solving for anti-derivatives is called anti-differentiation.

Anti-derivatives are not unique. If `F(x)` is an anti-derivative of `f(x)`, then so too is the function `F(x) + c` where `c` is any number. We write `int f(x) dx = F(x) + c` to denote the set of all anti-derivatives of `f(x)`. The number `c` is called the constant of integration. For example, since `d/dx (x^3) = 3x^2`, we can write `int 3x^2 dx = x^3 + c`.

Asymptote
A straight line is an asymptote of the function `y = f(x)` if graph of `y = f(x)` gets arbitrarily close to the straight line. An asymptote can be horizontal, vertical or oblique. For example, the line with equation `x = pi/2` is a vertical asymptote to the graph of `y = tan x`, and the line with equation `y = 0` is a horizontal asymptote to the graph of `y = 1/x`.

Bernoulli random variable
A Bernoulli random variable has two possible values, namely `0` and `1` . The parameter associated with such a random variable is the probability `p` of obtaining a `1`.

Bernoulli trial
A Bernoulli trial is a chance experiment with possible outcomes, typically labelled ‘success’ and failure’.

Binomial distribution
The expansion `(x + y)^n = x^n + ((n),(1))x^(n-1)y + ... + ((n),(r))x^(n-r)y^r + ... + y^n` is known as the binomial theorem. The numbers `((n),(r)) = (n!)/(r!(n - r)!) = (n xx (n - 1) xx ... xx (n - r + 1))/(r xx (r - 1) xx ... xx 2 xx 1)` are called binomial coefficients.

Central Limit Theorem
There are various forms of the Central Limit Theorem, a result of fundamental importance in statistics. For the purposes of this course, it can be expressed as follows: "If `bar X` is the mean of `n` independent values of random variable `X` which has a finite mean `mu` and a finite standard deviation `sigma`, then as `n -> oo`, the distribution of `(bar X - mu)/(sigma/sqrt n)` approaches the standard normal distribution.”

In the special case where `X` is a Bernoulli random variable with parameter `p`, `bar X` is the sample proportion `hat p`, `mu = p` and `sigma = sqrt (p(1 - p)`. In this case, the central limit theorem is a statement that as `n -> oo` the distribution of `(hat p - p)/sqrt ((hat p(1 - hat p))/n` approaches the standard normal distribution.

Chain rule
The chain rule relates the derivative of the composite of two functions to the functions and their derivatives. If `h(x) = f@g(x)`, then `(f@g)'(x) = f'(g(x))g'(x)`, and in Leibniz notation: `dz/dx = dz/dy dy/dx`.

Circular measure
A rotation, typically measured in radians or degrees.

Composition of functions
If `y = g(x)` and `z = f(y)` for functions `f` and `g`, then `z` is a composite function of `x`. We write `z = f@g(x) = f(g(x))`. For example, `z = sqrt (x^2 + 3)` expresses `z` as a composite of the functions `f(y) = sqrt y` and `g(x) = x^2 + 3`.

Concave up and concave down
A graph of `y = f(x)` is concave up at a point `P` if points on the graph near `P` lie above the tangent at `P`. The graph is concave down at `P` if points on the graph near `P` lie below the tangent at `P`.

Effect of linear change
The effects of linear changes of scale and origin on the mean and variance of a random variable are summarised as follows:
If `X` is a random variable and `Y = aX + b`, where `a` and `b` are constants, then `E(Y) = aE(X) + b` and  `Var(Y) = a^2 Var(X)`.

Euler’s number
Euler’s number `e` is an irrational number whose decimal expansion begins `e = 2.7182818284590452353602874713527...`.

It is the base of the natural logarithms, and can be defined in various ways including: `e = 1 + 1/(1!) + 1/(2!) + 1/(3!) + ...`  and  `e = lim_(n -> oo) (1 + 1/n)^n`.

Expected value
The expected value `E(X)` of a random variable `X` is a measure of the central tendency of its distribution.

If `X` is discrete, `E(X) = sum_i p_i x_i`, where the `x_i` are the possible values of `X` and `p_i = P(X = x_i)`.

If `X` is continuous, `E(x) = int_-oo^oo xp(x) dx`, where `p(x)` is the probability density function of `X`.

Function
A function `f` is a rule such that for each `x`-value there is only one corresponding `y`-value. This means that if `(a, b)` and `(a, c)` are ordered pairs, then `b = c`.

Gradient (Slope)
The gradient of the straight line passing through points `(x_1, y_1)` and `(x_2, y_2)` is the ratio `(y_2 - y_1)/(x_2 - x_1)`. Slope is a synonym for gradient.

Graph of a function
The graph of a function `f` is the set of all points `(x, y)` in Cartesian plane where `x` is in the domain of `f` and `y = f(x)`.

Index laws
The index laws are the rules: `a^x a^y = a^(x + y)`, `a^(-x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, and `(ab)^x = a^x b^x`, where `a`, `b`, `x` and `y` are real numbers.

Level of confidence
The level of confidence associated with a confidence interval for an unknown population parameter is the probability that a random confidence interval will contain the parameter.

Linearity property of the derivative
The linearity property of the derivative is summarised by the equations: `d/dx(ky) = k dy/dx` for any constant `k` and `d/dx (y_1 + y_2) = dy_1/dx + dy_2/dx`.

Local and global maximum and minimum
A stationary point on the graph `y = f(x)` of a differentiable function is a point where `f'(x) = 0`.

We say that `f(x_0)` is a local maximum of the function `f(x)` if `f(x) <= f(x_0)`for all values of `x` near `x_0`. 

We say that `f(x_0)` is a global maximum of the function `f(x)` if `(x) <= f(x_0)` for all values of `x` in the domain of `f`.

We say that `f(x_0)` is a local minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` near `x_0`.

We say that `f(x_0)` is a global minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` in the domain of `f`.

Margin of error
The margin of error of a confidence interval of the form `f - E < p < f + E` is `E`, the half-width of the confidence interval. It is the maximum difference between `f` and `p` if `p` is actually in the confidence interval.

Mean of a random variable
The mean of a random variable is another name for its expected value.

The variance `Var(X)` of a random variable `X` is a measure of the ‘spread’ of its distribution.

If `X` is discrete, `(X) = sum_i p_i(x_i - mu)^2`, where `mu = E(X)` is the expected value.

If `X` is continuous, `Var(X) = int_-oo^oo (x - mu)^2 p(x)dx`.

Non-routine problems
Problems solved using procedures not regularly encountered in learning activities.

Pascal’s triangle
Pascal’s triangle is a triangular arrangement of binomial coefficients. The `n^"th"` row consists of the binomial coefficients `((n),(r))`, for `0 <= r <= n`, each interior entry is the sum of the two entries above it, and sum of the entries in the `n^"th"` row is `2^n`.

Period of a function
The period of a function `f(x)` is the smallest positive number `p` with the property that `f(x + p) = f(x)` for all `x`. The functions `sin x` and `cos x` both have period `2pi`, and `tan x`  has period `pi`.

Point and interval estimates
In statistics estimation is the use of information derived from a sample to produce an estimate of an unknown probability or population parameter. If the estimate is a single number, this number is called a point estimate. An interval estimate is an interval derived from the sample that, in some sense, is likely to contain the parameter.

A simple example of a point estimate of the probability `p` of an event is the relative frequency `f` of the event in a large number of Bernoulli trials. An example of an interval estimate for `p` is a confidence interval centred on the relative frequency `f`.

Point of inflection
A point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.

Probability density function
The probability density function of a continuous random variable is a function that describes the relative likelihood that the random variable takes a particular value. Formally, if `p(x)` is the probability density of the continuous random variable `X`, then the probability that `X` takes a value in some interval `[a, b]` is given by `int_a^b p(x) dx`.

Probability distribution
The probability distribution of a discrete random variable is the set of probabilities for each of its possible values.

Product rule
The product rule relates the derivative of the product of two functions to the functions and their derivatives.

If`h(x) = f(x) g(x)`, then `h'(x) = f(x) g'(x) + f'(x) g(x)`, and in Leibniz notation: `d/dx(uv) = u (dv)/dx + (du)/dx v`.

Quadratic formula
If `ax^2 + bx + c = 0` with `a != 0`, then `x = (b +- sqrt(b^2 - 4ac))/(2a)`. This formula for the roots is called the quadratic formula.

Quantile
A quantile `t_alpha` for a continuous random variable `X` is defined by `P(X > t_alpha) = alpha`, where `0 < alpha < 1`.

The median `m` of `X` is the quantile corresponding to `alpha = 0.5 : P(X > m) = 0.5`.

Quotient rule
The quotient rule relates the derivative of the quotient of two functions to the functions and their derivatives
If `h(x) = f(x)/g(x)`, then `h'(x) = (g(x) f'(x) - f(x) g'(x))/g(x)^2`.

Radian measure
The radian measure `theta` of an angle in a sector of a circle is defined by `theta = l/r`, where `r` is the radius and `l` is the arc length. Thus, an angle whose degree measure is `180` has radian measures `pi`.

Random variable
A random variable is a numerical quantity, the value of which depends on the outcome of a chance experiment. For example, the proportion of heads observed in 100 tosses of a coin.

A discrete random variable is one which can only take a countable number of value, usually whole numbers.

A continuous random variable is one whose set of possible values are all of the real numbers in some interval.

Relative frequency
If an event `E` occurs `r` times in `n` trials of a chance experiment, the relative frequency of `E` is `r/n`.

Routine problems
Problems solved using procedures regularly encountered in learning activities.

Secant
A secant of the graph of a function is a straight line passing through two points on the graph. The line segment between the two points is called a chord.

Second derivative test
According to the second derivative test, if `f'(x) = 0`, then `f(x)` is a local maximum of `f` if `f''(x) < 0`, and `f(x)` is a local minimum if `f''(x) > 0`.

Sine and cosine functions
In the unit circle definition of cosine and sine, `cos theta` and `sin theta` are the `x` and `y` coordinates of the point on the unit circle corresponding to the angle `theta` measured as a rotation from the ray `OX`. If `theta` is measured in the counter-clockwise direction, then it is said to be positive; otherwise it is said to be negative.

Standard deviation of a random variable
The standard deviation of a random variable is the square root of its variance.

Tangent line
The tangent line (or simply the tangent) to a curve at a given point `P` can be described intuitively as the straight line that "just touches" the curve at that point. At the point where the tangent touches the curve, the curve has “the same direction” as the tangent line. In this sense it is the best straight-line approximation to the curve at the point.

The fundamental theorem of calculus
The fundamental theorem of calculus relates differentiation and definite integrals. It has two forms: `d/dx (int_a^x f(t) dt) = f(x)` and `int_a^b f'(x) dx = f(b) - f(a)`.

The linearity property of anti-differentiation
The linearity property of anti-differentiation is summarised by the equations: `int kf(x) dx = k int f(x) dx` for any constant `k`, and `int (f_1(x) + f_2(x)) dx = int f_1(x) dx + int f_2(x) dx` for any two functions `f_1(x)` and `f_2(x)`.

Similar equations describe the linearity property of definite integrals: `int_a^b kf(x)dx = kint_a^b f(x)dx` for any constant `k`, and `int_a^b(f_1(x) + f_2(x))dx = int_a^b f_1(x)dx + int_a^b f_2(x)dx` for any two functions `f_1(x)` and `f_2(x)`.

Uniform continuous random variable
A uniform continuous random variable `X` is one whose probability density function `p(x)` has constant value on the range of possible values of `X`. If the range of possible values is the interval `[a, b]`, then `p(x) = 1/(b - a)` if `a <= x <= b`.

Vertical line test
A relation between two real variables `x` and `y` is a function and `y = f(x)` for some function `f`, if and only if each vertical line, i.e. each line parallel to the `y`-axis, intersects the graph of the relation in at most one point. This test to determine whether a relation is, in fact, a function is known as the vertical line test.

LINE OF SIGHT – Mathematics Methods Level 4