THE ONTARIO CURRICULUM: PROPOSED REVISIONS

OCTOBER 2005

Mathematics: Grade 12

Advanced Functions

Side-by-Side

Original

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Revised

Course Code: MCB 4U Name: Advanced Functions and Introductory Calculus Grade: 12Program Area: Mathematics

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Revised: Advanced Functions

Strand: Advanced Functions

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Revised: Polynomial and Rational Functions

Section: Overall Expectations
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Unchanged: Overall Expectations
determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees; / determine the characteristics of a polynomial function of various degrees given its graph and make connections to its algebraic representation;
demonstrate facility in the algebraic manipulation of polynomials; / determine the characteristics of a polynomial function of various degrees, given its algebraic representation and make connections to its graph;
demonstrate an understanding of the nature of exponential growth and decay; / Moved to Grade 12 Advanced Functions - MCB 4U, Strand: Trigonometric, Exponential and Logarithmic Functions, Section: Overall Expectations
define and apply logarithmic functions; / Deleted.
demonstrate an understanding of the operation of the composition of functions. / Deleted.
Specific Expectations—Section: Investigating the Graphs of Polynomial Functions
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Revised: Investigating the Graphs of Polynomial and Rational Functions
determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions (e.g., determine the effect of the degree of a polynomial function on the shape of its graph; the effect of varying the coefficients in the polynomial function; the type and the number of x-intercepts; the behaviour near the x-intercepts; the end behaviours; the existence of symmetry); / Unchanged: determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions (e.g., determine the effect of the degree of a polynomial function on the shape of its graph; the effect of varying the coefficients in the polynomial function; the type and the number of x-intercepts; the behaviour near the x-intercepts; the end behaviours; the existence of symmetry);
describe the nature of change in polynomial functions of degree greater than two, using finite differences in tables of values; / Unchanged: describe the nature of change in polynomial functions of degree greater than two, using finite differences in tables of values;
compare the nature of change observed in polynomial functions of higher degree with that observed in linear and quadratic functions; / compare the nature of change observed in polynomial functions of higher degree with that observed in linear and quadratic functions (e.g., compare the graphs of f(x) = x^4 and f(x) = x²;
sketch the graph of a polynomial function whose equation is given in factored form; / Unchanged: sketch the graph of a polynomial function whose equation is given in factored form;
determine an equation to represent a given graph of a polynomial function, using methods appropriate to the situation (e.g., using the zeros of the function; using a trial-and-error process on a graphing calculator or graphing software; using finite differences). / Unchanged: determine an equation to represent a given graph of a polynomial function, using methods appropriate to the situation (e.g., using the zeros of the function; using a trial-and-error process on a graphing calculator or graphing software; using finite differences).
draw, using technology, the graph of rational functions (e.g., f(x) = 1/x, g(x) = 3/(x - 3), h(x) = (x - 2)/(x²-4) ) and identify through investigation, the key features of the graph (e.g., vertical vs. horizontal asymptotes, domain, range, positive/negative intervals, increasing/decreasing intervals);
sketch the graph of a rational function given its equation by considering the key features of the function f(x) = 1/x;
Specific Expectations—Section: Manipulating Algebraic Expressions
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Revised: Investigating the Algebra of Polynomial Functions
demonstrate an understanding of the remainder theorem and the factor theorem; / Unchanged: demonstrate an understanding of the remainder theorem and the factor theorem;
factor polynomial expressions of degree greater than two, using the factor theorem; / factor polynomial expressions of degree greater than two, using the factor theorem (e.g., x³ + 2x² - 1x - 2 and x^4 - 6x³ + 4x² + 6x - 5);
determine, by factoring, the real or complex roots of polynomial equations of degree greater than two; / determine, by factoring, the real roots of polynomial equations of degree greater than two (e.g., 2x³ - 3x² + 8x - 12 = 0) and verify graphically using technology;
determine the real roots of non-factorable polynomial equations by interpreting the graphs of the corresponding functions, using graphing calculators or graphing software; / determine the real roots of non-factorable polynomial equations (e.g., π x³- 4x² - 3x +π = 0) by interpreting the graphs of the corresponding functions, using graphing calculators or graphing software;
write the equation of a family of polynomial functions, given the real or complex zeros [e.g., a polynomial function having non-repeated zeros 5, 3, and 2 will be defined by the equation
f(x) = k(x5)(x + 3)(x + 2),
for k is an element of a set aleph]; / write the equation of a family of polynomial functions, given the zeros [e.g., a polynomial function having non-repeated zeros 5, -3, and -2 will be defined by the function f(x) = k(x- 5)(x + 3)(x + 2), for any real number k) and find the specific equation when given additional information;
verify, by investigation with technology (e.g., dynamic geometry software), that the output values of a polynomial function can only change sign at a zero;
describe intervals and distances, using absolute-value notation;
solve factorable polynomial inequalities; / solve linear and factorable polynomial inequalities, by determining intercepts and representing the solutions on number lines (e.g., x^4 - 5x² + 4 < 0);
solve non-factorable polynomial inequalities by graphing the corresponding functions, using graphing calculators or graphing software and identifying intervals above and below the x-axis; / solve non-factorable polynomial inequalities (e.g., x³ - x² + 3x - 9 ≥ 0) by graphing the corresponding functions (e.g., f(x) = x³ - x² + 3x - 9, using graphing calculators or graphing software and identifying intervals above and below the x-axis;
solve problems involving the abstract extensions of algorithms (e.g., a problem involving the nature of the roots of polynomial equations: If h and k are the roots of the equation
3x^2 + 28x - 20 = 0,
find the equation whose roots are h + k and hk; a problem involving the factor theorem: For what values of k does the function
f(x) = x^3 + 6x^2 + kx - 4
give the same remainder when divided by either x - 1 or x + 2?). / solve problems involving the abstract extensions of concepts related to polynomials (e.g., problems involving the nature of the roots of polynomial equations, problems involving the factor theorem); Sample problem: For what values of k does the function f(x) = x³ + 6x² + kx - 4 give the same remainder when divided by x - 1 and x + 2?
compare and describe, through investigation, the algebraic and graphical behaviour of even and odd polynomial functions (e.g., examining the values of the function for very large positive and negative values of x, the number of real roots, symmetry etc.) Sample problem: Under what conditions will an even function have an even number of zeros?
solve equations and inequalities involving simple rational functions, graphically, and algebraically, using zeros, asymptotes and the values of the function between these intervals;
Specific Expectations—Section: Understanding the Nature of Exponential Growth and Decay
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Deleted.
identify, through investigations, using graphing calculators or graphing software, the key properties of exponential functions of the form
a^x (a > 0, a != 1)
and their graphs (e.g., the domain is the set of the real numbers; the range is the set of the positive real numbers; the function either increases or decreases throughout its domain; the graph has the x-axis as an asymptote and has y-intercept = 1);
describe the graphical implications of changes in the parameters a, b, and c in the equation
y = ca^x + b;
compare the rates of change of the graphs of exponential and non-exponential functions (e.g., those with equations
y = 2x,
y = x^2,
y = x^(1/2), and
y = 2^x);
describe the significance of exponential growth or decay within the context of applications represented by various mathematical models (e.g., tables of values, graphs);
pose and solve problems related to models of exponential functions drawn from a variety of applications, and communicate the solutions with clarity and justification.
Specific Expectations—Section: Defining and Applying Logarithmic Functions
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Deleted.
define the logarithmic function
log to the base a of x (a > 1)
as the inverse of the exponential function a^x, and compare the properties of the two functions;
express logarithmic equations in exponential form, and vice versa;
simplify and evaluate expressions containing logarithms;
solve exponential and logarithmic equations, using the laws of logarithms;
solve simple problems involving logarithmic scales (e.g., the Richter scale, the pH scale, the decibel scale).
Specific Expectations—Section: Understanding the Composition of Functions
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Unchanged: Understanding the Composition of Functions
Moved from this location.
identify composition as an operation in which two functions are applied in succession; / Moved to Grade 12 Advanced Functions - MCB 4U, Strand: Trigonometric, Exponential and Logarithmic Functions, Section: Connecting Functions, Moved w Section
demonstrate an understanding that the composition of two functions exists only when the range of the first function overlaps the domain of the second; / Moved to Grade 12 Advanced Functions - MCB 4U, Strand: Trigonometric, Exponential and Logarithmic Functions, Section: Connecting Functions, Moved w Section
determine the composition of two functions expressed in function notation; / Deleted.
decompose a given composite function into its constituent parts; / Deleted.
describe the effect of the composition of inverse functions [i.e.,
f(f^(-1)(x)) = x]. / Moved to Grade 12 Advanced Functions - MCB 4U, Strand: Trigonometric, Exponential and Logarithmic Functions, Section: Connecting Functions, Moved w Section

Strand: New strand added.

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New: Trigonometric, Exponential and Logarithmic Functions

Specific Expectations—Section: New section added.
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Revised: Overall Expectations
extend an understanding of exponential functions and the related logarithmic functions to solve problems involving exponential growth and decay;
extend an understanding of trigonometric functions using radian measure and solve related problems;
(Expectation moved here; formerly AFV.03)
demonstrate an understanding of the nature of exponential growth and decay; / consolidate their understanding of the characteristics of functions by considering compound functions and the composition of functions
Specific Expectations—Section: New section added.
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Revised: Exponential and Logarithmic Functions

solve exponential equations by finding a common base (e.g., 4^x = 8^(x+3), 2^(x+2) - 2^x = 12).
evaluate numerical expressions involving logarithms (e.g., log 10(29), log3(25), log10(400) - log(10) 4), using a calculator,;
determine, through investigation, the laws of logarithms and use them to simplify and evaluate logarithmic expressions;
write a logarithmic statement in exponential form, and vice versa;
solve exponential and logarithmic equations using the laws of logarithms
define the logarithmic function f(x) = log b( x )(b > 0, b/=1) as the inverse of the exponential function f(x) = b^x;
compare the properties of the exponential and logarithmic functions;
pose and solve problems related to models of exponential and logarithmic functions drawn from a variety of applications (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale)

Specific Expectations—Section: New section added.

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Revised: Trigonometric Functions

define radian measure and develop the relationship between radian and degree measure;
represent, in applications, radian measure in exact form, as an expression involving π (e.g., π/3, 2π) and in approximate form as a rational number (e.g. 1.05)
determine the exact values of the sine, cosine, and tangent of the special angles 0, π/6, π/4, π/3, π/2, and their multiples;
demonstrate facility in the use of radian measure in graphing (e.g., f(x) = cos(x), g(x)=2sin(x+ π/3)
demonstrate facility in the use of the reciprocal trigonometric ratios (i.e., cosecant, secant and tangent)
sketch the graph of the tangent function and the reciprocal trigonometric functions and identify the key features of their graphs (e.g., state the domain, range, and period and identify and explain the occurrence of asymptotes)
demonstrate an understanding of the development of the compound angle and double angle formulae and the formulae to determine exact trigonometric values (e.g., determine the exact value of sin(pi/12));
solve, with and without graphing technology, linear and quadratic trigonometric equations between 0 and 2π, and over R (the Real numbers);
determine, through investigation using graphing technology, whether or not two trigonometric expressions are equivalent;
prove trigonometric identities using a variety of relationships, including the reciprocal relationships and the compound angle formulae;
pose and solve problems related to models of trigonometric functions drawn from a variety of applications (e.g., tides, length of day, oscillating spring) with justification, with and without technology;

Specific Expectations—Section: Understanding the Composition of Functions

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Revised: Connecting Functions

Moved to this location.
describe, through investigation using a variety of tools and strategies, some of the properties of compound functions (e.g., f(x) = x sin x, g(x) = x² +2^x);
(Expectation moved here; formerly AF5.01)
identify composition as an operation in which two functions are applied in succession; / demonstrate an understanding of composition as an operation in which two functions are applied in succession and determine the composition of two functions expressed in functional notation
(Expectation moved here; formerly AF5.02)
demonstrate an understanding that the composition of two functions exists only when the range of the first function overlaps the domain of the second; / demonstrate an understanding of the domain and range of the composition of two functions (i.e., f(g(x)) is defined for those x for which g(x) is defined and included in the domain of f(x))
(Expectation moved here; formerly AF5.05)
describe the effect of the composition of inverse functions [i.e.,
f(f^(-1)(x)) = x]. / describe, using a variety of representations and an understanding of the inverse as a reverse process, the effect of the composition of inverse functions [i.e., f(f^(-1)(x)) = x].
compare and contrast, through investigation, the characteristics (e.g., symmetry, asymptotes, intercepts, domain and range, increasing/decreasing, critical points) of functions (i.e., linear, quadratic, trigonometric, exponential, logarithmic, polynomial, rational) using a variety of representations (e.g., tables of values, function machines, graphs and algebraic representations);

Strand: Underlying Concepts of Calculus

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Revised: Rates of Change

Section: Overall Expectations

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Unchanged: Overall Expectations

determine and interpret the average and instantaneous rates of change of given functions
(Expectation moved here; formerly CCV.03)
demonstrate an understanding of the relationship between the derivative of a function and the key features of its graph. / demonstrate an understanding of the relationship between the shape of a graph and the rate of change of the dependent variable.
determine and interpret the rates of change of functions drawn from the natural and social sciences; / Unchanged: determine and interpret the rates of change of functions drawn from the natural and social sciences;
demonstrate an understanding of the graphical definition of the derivative of a function;
demonstrate an understanding of the relationship between the derivative of a function and the key features of its graph. / Moved to Grade 12 Advanced Functions - MCB 4U, Strand: Rates of Change, Section: Overall Expectations

Specific Expectations—Section: Understanding Rates of Change

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Unchanged: Understanding Rates of Change

pose problems and formulate hypotheses regarding rates of change within applications drawn from the natural and social sciences; / pose problems and formulate hypotheses regarding rates of change within applications drawn from mathematics (e.g., rate of change of the area of a circle as the radius increases) and from the real world (e.g., inflation rates, cycling up a hill, infection rates) Sample Problem: Given that the bacteria count in a sample is 1 000 000 at 1:00 pm, and 250 000 at 3:00 pm, pose and solve a problem involving the rate of change of the bacterial population.)
calculate and interpret average rates of change from various models (e.g., equations, tables of values, graphs) of functions drawn from the natural and social sciences; / calculate and interpret average rates of change from various representations (e.g., equations, tables of values, graphs) of functions drawn from the natural and social sciences; Enter Your Revision:
estimate and interpret instantaneous rates of change from various models (e.g., equations, tables of values, graphs) of functions drawn from the natural and social sciences; / estimate instantaneous rates of change given various representations (e.g., algebraic, graphical);
interpret the meaning of the instantaneous rates of change; Sample Problem: If the instantaneous rate of change is given by a speedometer as 60 km/h, interpret the meaning);
explain the difference between average and instantaneous rates of change within applications and in general; / demonstrate an understanding of the difference between average and instantaneous rates of change using relevant applications (e.g., for a given average velocity over an interval, there must be at least one point in time in that interval, where the average is the instantaneous velocity) Sample Problem: How can you determine the average rate of change and the instantaneous rate of change from a table of values?
make inferences from models of applications and compare the inferences with the original hypotheses regarding rates of change. / Unchanged: make inferences from models of applications and compare the inferences with the original hypotheses regarding rates of change.

Specific Expectations—Section: Understanding the Graphical Definition of the Derivative

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Revised: Interpreting Rates of Change on a Graph

demonstrate an understanding that the slope of a secant on a curve represents the average rate of change of the function over an interval, and that the slope of the tangent to a curve at a point represents the instantaneous rate of change of the function at that point; / Unchanged: demonstrate an understanding that the slope of a secant on a curve represents the average rate of change of the function over an interval, and that the slope of the tangent to a curve at a point represents the instantaneous rate of change of the function at that point;
demonstrate an understanding that the slope of the tangent to a curve at a point is the limiting value of the slopes of a sequence of secants; / demonstrate, through investigation, an understanding that the slope of the tangent to a curve at a point can be approximated by the slope of a secant;
demonstrate an understanding that the instantaneous rate of change of a function at a point is the limiting value of a sequence of average rates of change; / demonstrate an understanding that the instantaneous rate of change of a function at a point can be approximated by average rates of change;
demonstrate an understanding that the derivative of a function at a point is the instantaneous rate of change or the slope of the tangent to the graph of the function at that point. / determine when the rate of change is increasing or decreasing from the shape of the graph of a function;
demonstrate an understanding of the concept of acceleration (e.g., rate of change of velocity, accelerating cost), graphically, numerically (i.e., second differences), algebraically, verbally;
(Expectation moved here; formerly CC3.03)
sketch, by hand, the graph of the derivative of a given graph. / sketch, by hand, the graph of the rate of change of a function, (i.e. using the slopes of the tangents), given the graph of the function;
sketch the graph of a function, given the graph of its rate of change (e.g., given a velocity-time graph, sketch the position-time graph );
(Expectation moved here; formerly CC3.01)
describe the key features of a given graph of a function, including intervals of increase and decrease, critical points, points of inflection, and intervals of concavity; / describe the key features of a given graph of a function, including intervals of increase and decrease, local and absolute extrema, endpoints, points of inflection, and intervals of concavity;
(Expectation moved here; formerly CC3.02) / Unchanged: identify the nature of the rate of change of a given function, and the rate of change of the rate of change, as they relate to the key features of the graph of that function;

Specific Expectations—Section: Using Calculus Techniques to Analyse Models of Functions