Advanced Functions

In this unit you will meet the logarithm, the exponent you are looking for when you ask "2 to what power gives 50?" You will evaluate logarithms by hand and with a calculator (using the change of base formula), switch between exponential and logarithmic form, and prove the product, quotient and power laws of logarithms from the exponent laws. You will graph y = log_b x, see that it is the inverse of y = bˣ (a reflection in the line y = x), and transform it with y = a log (k(x − d)) + c. Then you will solve exponential equations by finding a common base or by taking the logarithm of both sides, solve logarithmic equations and reject extraneous roots, and use all of it on real problems: pH, decibels, earthquake magnitude, half-life, doubling time, compound interest and cooling.

Remember from Grade 11: in Functions (MCR3U) you graphed f(x) = bˣ, used the exponent laws with rational exponents, transformed functions with y = af(k(x − d)) + c, found inverses by swapping x and y, and modelled growth and decay with exponential functions (solving for time by graphing or trial). This unit gives you the algebra to solve those problems exactly.

You will need a scientific calculator with LOG and 10ˣ keys. Free graphing technology such as Desmos is helpful for investigating graphs, but every answer can be found by hand or with a scientific calculator. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: the unit Radian Measure and Trigonometric Functions begins the trigonometry strand of this course. Logarithms return in Calculus and Vectors (MCV4U), where you will meet the natural logarithm ln x.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand A. Exponential and Logarithmic Functions: A1.1 to A1.4 (evaluating logarithmic expressions), A2.1 to A2.4 (connecting graphs and equations of logarithmic functions) and A3.1 to A3.4 (solving exponential and logarithmic equations). The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

Mathematicians, scientists and engineers measure angles in radians, and in this unit you will learn why. A radian measures an angle by the arc it cuts off, in radii, so a full turn is 2π and the input of sin and cos becomes an ordinary real number. You will convert between radians and degrees, find arc lengths, and find the six trigonometric ratios of radian angles both with a calculator and exactly from the unit circle (π/6, π/4, π/3 and their relatives, including cosecant, secant and cotangent).

Then you will graph all six trigonometric functions in radians: y = sin x and y = cos x, the tangent function with its vertical asymptotes, and the reciprocal functions csc x, sec x and cot x. You will read the amplitude, period (2π/|k|) and phase shift from y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c, sketch these functions by hand, write equations from graphs and descriptions, and finally use sinusoidal models of Ferris wheels, tides, hours of daylight and springs to answer real questions: values, times, how long, and how fast things change.

Remember from Grade 11: in Functions (MCR3U), the units Trigonometric Ratios and Laws and Sinusoidal Functions covered special angles, CAST, the reciprocal ratios and transformed sine and cosine graphs, all in degrees. This unit redoes that work in radians and adds the tangent and reciprocal graphs. You will need a scientific calculator (set to radian mode). A free online graphing tool helps with the investigations, but every question can be answered by hand or with a scientific calculator. Each of the 13 lessons has worked examples, a hands-on activity, practice with full answers and a short quiz, and the unit ends with a review.

Coming next: Trigonometric Identities and Equations (the rest of strand B) proves identities and solves trigonometric equations in radians. Rates of Change builds on the rate-of-change ideas in the last lesson.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand B. Trigonometric Functions: overall expectation B1 (radian measure, B1.1 to B1.4) and B2 (connecting graphs and equations of trigonometric functions, B2.1 to B2.7). Overall expectation B3 is taught in Trigonometric Identities and Equations. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will learn to see when two trigonometric expressions are really the same, and use that skill to prove identities and solve equations. You will start with equivalent expressions: the cofunction identities such as sin x = cos(π/2 − x), and expressions that come from shifting graphs or using the symmetry of the unit circle, such as cos(x + π/2) = −sin x. Then you will follow the development of the compound angle formulas for cos(a ± b), sin(a ± b) and tan(a ± b), verify them with numbers, and use them and the double angle formulas to find exact values like cos(π/12) = (√6 + √2)/4.

The middle of the unit is about identities: equations that are true for every value in their domain. You will use a counterexample to show that an equation is not an identity, check identities with graphing technology, and write careful proofs using the reciprocal, quotient, Pythagorean, compound angle and double angle identities. The last part is about solving: linear and quadratic trigonometric equations in radians for 0 ≤ x ≤ 2π, equations with angles like 2x or x − π/4, equations that need an identity first, and real problems about Ferris wheels, tides in the Bay of Fundy and hours of daylight.

Remember from Grade 11: in Functions (MCR3U), the unit Trigonometric Ratios, Identities and Laws gave you the special-angle values, the CAST rule, two angles with the same ratio, the reciprocal ratios and proofs with the Pythagorean, quotient and reciprocal identities, all in degrees. Remember from this course: the unit Radian Measure and Trigonometric Functions introduced radians, exact values for angles like π/6 and 3π/4, and the graphs of the trigonometric functions. Keep your calculator in radian mode throughout. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: the unit Polynomial Functions and Equations begins strand C.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand B. Trigonometric Functions, overall expectation B3 (solve problems involving trigonometric equations and prove trigonometric identities): specific expectations B3.1 to B3.4. Overall expectations B1 and B2 are taught in Radian Measure and Trigonometric Functions. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you study polynomial functions of any degree: cubics, quartics and beyond. You will decide what is (and is not) a polynomial, use finite differences to find the degree from a table, and describe the key features of polynomial graphs: end behaviour, turning points, zeros and their order, domain and range, and even and odd symmetry. You will tell polynomials apart from exponential and sinusoidal functions, sketch graphs from factored form, transform y = x³ and y = x⁴, and write equations from families of polynomials and other conditions.

The second half is algebra. You will divide polynomials (long and synthetic division), use the remainder theorem and the factor theorem, and factor polynomials of degree up to 4, including the sum and difference of cubes, grouping and quadratic form. You will solve polynomial equations exactly and by estimation, solve real problems with polynomial models, and solve polynomial inequalities graphically and algebraically, writing answers in interval notation.

Remember from Grade 11: in Functions (MCR3U) you worked with function notation, transformations y = af(k(x − d)) + c, families of quadratics with the same zeros, and multiplying polynomials. This unit builds directly on those skills. You need a scientific calculator; a free graphing tool such as Desmos or GeoGebra is helpful for checking. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: the unit Rational Functions and Equations divides one polynomial by another and studies asymptotes and rational equations and inequalities.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand C. Polynomial and Rational Functions: overall expectation C1 (specific expectations C1.1 to C1.9), the polynomial parts of C3 (C3.1 to C3.4, and C3.7 for polynomial models) and C4 (C4.1 to C4.3 for polynomial and linear inequalities). Rational functions (C2, C3.5 to C3.7 and rational inequalities) are taught in Rational Functions and Equations. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

This unit is about functions with a variable in the denominator: rational functions. You will start with reciprocals: y = 1/f(x) for linear and quadratic functions f. You will see how every feature of the graph comes from the original function. Zeros become vertical asymptotes, large values become small ones, values of ±1 stay put, and the vertex of a parabola becomes a turning point. Next you will analyse functions of the form y = (ax + b)/(cx + d): their asymptotes, intercepts, holes, increasing and decreasing intervals, and the transformed form y = k/(x − h) + q. Then you will put it all together in a five-step strategy for sketching any simple rational function by hand.

The second half turns to equations and inequalities. You will connect the roots of a rational equation to the x-intercepts of a graph, solve rational equations algebraically while rejecting extraneous roots, rearrange formulas such as the thin-lens equation, and model work, travel, average cost, concentration, light intensity and growth that levels off. Finally, you will solve rational inequalities with graphs and with sign charts, and learn why you must never multiply an inequality by an expression whose sign you do not know.

Remember from Grade 11: in Functions (MCR3U) you met the parent function y = 1/x and translated it (unit Functions and Equivalent Expressions), simplified and combined rational expressions with restrictions (same unit), and found zeros, vertices and discriminants of quadratics (unit Quadratic Functions). This unit builds directly on all three. A scientific calculator is enough for every question; a graphing calculator or a free online graphing tool is useful for checking.

Related units in this course: Polynomial Functions and Equations covers polynomial graphs, the factor and remainder theorems, and polynomial inequalities (expectations C1, C3.1 to C3.4 and C4.3). Combining Functions and Modelling compares all the function families of the course and combines them.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand C. Polynomial and Rational Functions: overall expectation C2 (specific expectations C2.1 to C2.3), the rational parts of C3 (C3.5 to C3.7) and of C4 (C4.1 and C4.2). The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will study how fast things change. You will start by describing rates of change in words, tables, graphs and equations, telling zero, constant and changing rates apart, and sketching graphs from stories (filling containers, walking in front of a motion sensor, a snowstorm). Then you will calculate average rates of change as slopes of secant lines for every kind of function in this course, with real models from biology, finance, tides and design.

The heart of the unit is the instantaneous rate of change: the rate at a single moment, like a speedometer reading. You will see that it is the slope of the tangent line, and you will find it three ways: by shrinking the interval (preceding, following and centred), by watching secants approach the tangent, and exactly, with the difference quotient (f(a + h) − f(a)) ÷ h. You will discover the rate-of-change patterns of polynomial, exponential, logarithmic, sinusoidal and rational functions, read rates of change straight from graphs, sketch slope graphs, and solve multi-step problems.

Remember from Grade 11: in Functions (MCR3U) you used first differences, met secant and tangent lines to a parabola in Quadratic Functions, and worked with exponential models. If you took Physics (SPH3U), you found average and instantaneous velocity from secants and tangents in Kinematics. This unit also uses the earlier MHF4U units: Polynomial Functions and Equations, Rational Functions and Equations, Exponential and Logarithmic Functions, Radian Measure and Trigonometric Functions and Trigonometric Identities and Equations. You will need a scientific calculator (set to radian mode for sinusoidal functions); a free graphing tool such as Desmos or GeoGebra is helpful but not required. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: the unit Combining Functions and Modelling completes strand D. After MHF4U, Grade 12 Calculus and Vectors (MCV4U) begins with the unit Limits and Rates of Change, which turns the ideas here (letting h approach 0, slope graphs) into limits and derivatives.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand D. Characteristics of Functions, overall expectation D1 (understanding rates of change): specific expectations D1.1 to D1.9. Overall expectations D2 and D3 are taught in Combining Functions and Modelling. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

This is the capstone unit of Advanced Functions. You will treat functions the way you treat numbers: add, subtract, multiply and divide them, and find the domain, range, maximum and minimum points and zeros of the results. You will test combined functions for even and odd symmetry and for increasing and decreasing behaviour, and you will meet real combinations such as a damped spring (an exponential times a cosine), a phone keypad tone (a sum of two sines) and profit (revenue minus cost).

The middle of the unit is about composition, feeding one function into another. You will compose functions from tables, graphs and equations, find the domain and range of f(g(x)), see why f(g(x)) is usually not g(f(x)), solve real problems with composite models, decompose functions, show that f(f⁻¹(x)) = x, and recognize every transformation as a composition with a linear function.

The unit ends by pulling the whole course together: you will compare all the function families (polynomial, rational, exponential, logarithmic and trigonometric), solve equations and inequalities that algebra cannot reach using graphs and numerical methods such as bisection, and build, use and interpret models of real situations.

Remember from earlier units: this unit uses everything from Exponential and Logarithmic Functions, Radian Measure and Trigonometric Functions, Trigonometric Identities and Equations, Polynomial Functions and Equations, Rational Functions and Equations and Rates of Change, and the Grade 11 ideas of function notation, inverses and transformations from Functions (MCR3U). You will need a scientific calculator (with radian mode for trigonometric work). A free online grapher such as Desmos is useful for checking, but every answer can be found by hand.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Advanced Functions, Grade 12, University Preparation (MHF4U), strand D. Characteristics of Functions, overall expectations D2 (combining functions) and D3 (using function models to solve problems): specific expectations D2.1 to D2.8 and D3.1 to D3.3. Overall expectation D1 (rates of change) is taught in the unit Rates of Change. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.