Functions

This unit opens the course by asking a simple question with big consequences: what exactly is a function? You will tell functions apart from other relations (using tables, mapping diagrams, graphs and equations, and the vertical line test), write them in function notation, and describe their domain and range in set notation, including the limits that real situations put on them. You will get to know four parent functions, f(x) = x, x², √x and 1/x, and find inverses: undoing a process, swapping coordinates, reflecting in the line y = x, and solving for the inverse of linear and quadratic functions (restricting the domain when needed). Then you will transform the parent functions with y = af(k(x − d)) + c: translations, stretches, compressions and reflections, sketched by hand with mapping rules.

The second half builds the algebra this course depends on: adding, subtracting and multiplying polynomials, simplifying, multiplying, dividing, adding and subtracting rational expressions with their restrictions, and deciding whether two expressions are equivalent, including a short introduction to equivalent radical forms using √(ab) = √a × √b. (Radicals get a full lesson, "Exact Answers: Simplifying and Operating with Radicals", in the Quadratic Functions unit, where they are needed for exact zeros.)

Remember from earlier grades: in Grade 9 you compared linear and non-linear relations, graphed xy = k, worked with the real number system, collected like terms and multiplied binomials. In Grade 10 you factored quadratic expressions, completed the square and transformed y = x² into y = a(x − h)² + k. This unit builds on all of these. The next unit, Quadratic Functions, uses them to find zeros, maximum and minimum values, and intersections. You will need grid paper, a ruler and a scientific calculator; a free graphing tool is helpful for investigating but not required. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Functions, Grade 11, University Preparation (MCR3U), strand A. Characteristics of Functions: overall expectation A1 (specific expectations A1.1 to A1.9, representing functions, inverses and transformations) and overall expectation A3 (A3.1 to A3.4, determining equivalent algebraic expressions; A3.2 is introduced briefly here and taught in full in the Quadratic Functions unit). Overall expectation A2 (quadratic functions) is taught in the Quadratic Functions unit. 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 quadratic functions in depth. You will write them in standard, vertex and factored form using function notation, and find their maximum or minimum values by completing the square (now with fractions and decimals), by partial factoring and with x = −b/(2a). You will learn to simplify radicals so you can give exact zeros from the quadratic formula, and you will decide how many zeros a function has by inspecting its graph, by factoring and with the discriminant, including which transformations change that number. You will investigate families of quadratic functions with the same zeros and find the equation of a quadratic from its zeros and one point, even when the zeros are irrational. Finally you will solve real-world problems (revenue, profit, area, projectiles) and find where a line meets a parabola, graphically and algebraically, including tangent lines.

Remember from Grade 10 (MPM2D): you already graphed parabolas, found vertices, zeros and axes of symmetry, expanded and factored, completed the square when half of b was a whole number, used the quadratic formula and the discriminant, and solved linear systems. This unit builds on all of that. The sibling unit Functions and Equivalent Expressions covers function notation, domain and range, inverses and transformations, which you will use here. A scientific calculator is enough for every question; graphing technology is helpful for checking. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions, Grade 11, University Preparation (MCR3U), strand A. Characteristics of Functions, overall expectation A2 (determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications): A2.1 to A2.5. Lesson 4 also teaches A3.2 (simplifying radicals and radical expressions), which the exact answers in this unit depend on. 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 functions where the variable is in the exponent, such as f(x) = 2ˣ. You will start by graphing y = aˣ and explaining why it is a function. Then you will find out what rational exponents such as 8^(2/3) mean, and use the exponent laws to simplify expressions with integer and rational exponents. You will describe the key properties of exponential graphs (domain, range, intercepts, asymptote, increasing or decreasing), tell exponential functions apart from linear and quadratic ones, and transform their graphs using y = af(k(x − d)) + c. You will see that one function can be written with different bases (9ˣ = 3^(2x)), find equations from graphs and properties, collect data that grows or decays exponentially, and solve real problems about radioactive decay, population growth, cooling, bouncing balls and compound interest.

Remember from earlier grades: in Grade 9 you learned the exponent laws and used compound interest; in Grade 10 you compared y = x² with y = 2ˣ, met zero and negative exponents in the graph of y = 2ˣ, and transformed parabolas with y = a(x − h)² + k. This unit builds on all of these. A scientific calculator is needed; free graphing technology such as Desmos is helpful but every answer can be found by hand or with a calculator. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions, Grade 11, University Preparation (MCR3U), strand B. Exponential Functions: B1.1 to B1.4 (representing exponential functions; rational exponents), B2.1 to B2.5 (connecting graphs and equations of exponential functions) and B3.1 to B3.3 (solving problems involving exponential functions). The mathematical process expectations are woven through every lesson.

In this unit you will study discrete functions: functions whose inputs are the positions 1, 2, 3, … of a list. You will see a sequence as a function on the natural numbers and tell discrete functions from continuous ones. You will describe sequences in words, with flow charts, with recursion formulas, with function notation and with a formula for the nth term, and learn what each representation shows. You will investigate the Fibonacci sequence and related sequences, discover the patterns hidden in Pascal's triangle, and use the triangle to expand binomials such as (2x − 3)⁴. Then you will meet the two great families of sequences, arithmetic (add the same number) and geometric (multiply by the same number), develop formulas for their general terms and for the sums of their series, and use them to solve real problems: arena seating, bouncing balls, phone trees, falling objects and growing populations.

Remember from earlier grades: in Grade 9 you met the natural numbers, first differences, and sequences that approach a limit; in Grade 10 you used second differences and expanded (a + b)². Earlier in this course you used function notation (Functions and Equivalent Expressions) and studied exponential functions (Exponential Functions). The next unit, Financial Applications, uses geometric sequences and series for compound interest and annuities. You will need a scientific calculator. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions, Grade 11, University Preparation (MCR3U), strand C. Discrete Functions: C1.1 to C1.6 (representing sequences) and C2.1 to C2.4 (investigating arithmetic and geometric sequences and series). 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 use the mathematics of the unit Sequences and Series and of Exponential Functions to understand money. You will see why simple interest grows like an arithmetic sequence (linear growth) and why compound interest grows like a geometric sequence (exponential growth). You will use the compound interest formula A = P(1 + i)ⁿ with any compounding period, and solve it for the future value, the present value, the interest rate (with rational exponents) and the number of compounding periods (with guess and check, graphs and a TVM Solver). Then you will meet annuities: equal payments at equal intervals. You will show that an annuity's value is the sum of a geometric series, and use that to find the future value, the present value and the regular payment of savings plans and loans. Finally you will use spreadsheets and a TVM Solver to ask "what if?" about payments, rates, frequencies and terms, build an amortization table, and compare the total interest on a loan with the amount borrowed.

Remember from earlier grades: in Grade 8 you met simple interest (I = Prt) and compound interest; in Grade 9 you compared compounding periods, the cost of borrowing and spreadsheet amortization tables. You will need a scientific calculator (with a power key such as ^ or xʸ) and, for some activities, a free spreadsheet app. All rates and prices in this unit are examples; real rates change often. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

In this unit you will extend trigonometry far beyond the right triangles of Grade 10. You will start with the special triangles (45-45-90 and 30-60-90) and find exact values for 0°, 30°, 45°, 60° and 90°. Then you will put angles in standard position on a coordinate grid, define sine, cosine and tangent for every angle from 0° to 360° using a point (x, y) and its distance r, and use the CAST rule and related acute angles to find exact values such as cos 225° = −√2/2. You will learn why an equation like sin θ = 0.4 has two answers, meet the reciprocal ratios cosecant, secant and cotangent, and prove trigonometric identities with the Pythagorean, quotient and reciprocal identities.

The second half returns to triangles. You will use the sine law and cosine law in obtuse triangles, master the ambiguous case of the sine law (no triangle, one or two), and pose and solve problems in two dimensions (navigation with true bearings, towers, lots) and in three dimensions (rooms, boxes, pyramids, towers seen from different directions).

Remember from Grade 10: in Principles of Mathematics (MPM2D) you used SOH CAH TOA and the Pythagorean theorem in right triangles, angles of elevation and depression, and the sine and cosine laws in acute triangles. Everything here builds on that. You will need a scientific calculator set to degree mode. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: in the unit Sinusoidal Functions you will graph sine and cosine as functions and use them to model repeating patterns such as tides and hours of daylight.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Functions, Grade 11, University Preparation (MCR3U), strand D. Trigonometric Functions, overall expectation D1 (determining and applying trigonometric ratios): specific expectations D1.1 to D1.7. Overall expectations D2 and D3 are taught in Sinusoidal 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.

Ferris wheels, tides, hours of daylight, a swinging pendulum, the electricity in your walls: all of them go up and down in cycles. In this unit you will learn the mathematics of things that repeat. You will start with periodic functions and the words that describe them (cycle, period, amplitude, axis) and use the period to predict future behaviour. Then you will turn the sine and cosine ratios from the unit Trigonometric Ratios and Laws into the sine and cosine functions, learn their graphs, and discover how the four parameters of y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c stretch, reflect and move them. You will sketch any sinusoidal function by hand, write equations from graphs and properties, and finally put the functions to work: collecting and modelling data, modelling situations that do not involve angles at all, predicting what happens when conditions change, and solving real problems.

Remember from earlier: in Grade 10 (MPM2D) you transformed parabolas with y = a(x − h)² + k and used the primary trigonometric ratios in right triangles; in the unit Trigonometric Ratios and Laws you found sine and cosine for angles from 0° to 360° and found two angles with the same ratio. You will need a scientific calculator in degree mode; a free online graphing tool is helpful for the investigations but every question can be answered by hand. Each of the 13 lessons has worked examples, a hands-on activity, lots of practice and a short quiz, and the unit ends with a review.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Functions, Grade 11, University Preparation (MCR3U), strand D. Trigonometric Functions: D2.1 to D2.8 (connecting graphs and equations of sinusoidal functions) and D3.1 to D3.5 (solving problems involving sinusoidal 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.