Functions and Applications

This unit is the algebra toolkit for the whole quadratic strand of Functions and Applications. You will start by posing and solving problems from tables and graphs of quadratic relations. Then you will represent real situations (frames, borders, gutters, prices and revenue) with quadratic expressions, expand and simplify them, and factor them in every form the course requires: common factors, trinomials x² + bx + c and ax² + bx + c, differences of squares and perfect square trinomials. You will solve quadratic equations by factoring, discover how the factors give the x-intercepts of a graph, follow the development of the quadratic formula by completing the square, and use the formula with a calculator. The discriminant will tell you how many real roots to expect, and you will learn to choose between graphing, factoring and the formula, weighing the advantages and disadvantages of each. The unit ends with real problems that put every step together.

Remember from Grade 10: you met second differences, parabolas, expanding, factoring, solving by factoring and the quadratic formula. Grade 11 revisits each idea quickly, then goes further: messier equations such as 100x² = 115x + 35, decimal coefficients from real situations, finding unknown coefficients with the discriminant, and deciding which method is best. The companion unit, Quadratic Functions, uses these skills to study function notation, domain and range, transformations, vertex form and maximum and minimum problems.

You will need a scientific calculator. A free graphing tool (Desmos or GeoGebra on a computer or phone) is helpful for checking, but every answer can be found by hand or with a scientific calculator. Every lesson has worked examples, at least two diagrams, a hands-on activity, plenty of practice with full solutions and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions and Applications, Grade 11, University/College Preparation (MCF3M), strand A. Quadratic Functions, overall expectation A1, Solving Quadratic Equations (expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph): specific expectations A1.1 to A1.8. 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 look at quadratics as functions. You will start with the big idea of Grade 11 math: what makes a relation a function, and how to use function notation such as f(3) = 7. You will describe the domain and range of linear and quadratic functions in set notation, and explain the restrictions a real situation puts on them (a snowball can't fly before it is thrown).

Next you will investigate how a, h and k transform the parent function f(x) = x², and sketch g(x) = a(x − h)² + k in under a minute. You will move between the three forms of a quadratic function, standard, vertex and factored, by expanding, completing the square and factoring, verify that the forms are equivalent, describe what each form shows, and sketch from any of them.

In the last part you will collect data that curves like a parabola, find a quadratic model for it by hand and compare it with a regression from technology, and use quadratic functions to solve problems about heights, arches, areas, revenue and profit.

Remember from Grade 10: you graphed quadratic relations, used second differences, met vertex form and transformations, completed the square, and solved problems about thrown objects and areas. This unit builds those ideas into the language of functions. The companion unit Quadratic Expressions and Equations covers expanding, factoring, solving quadratic equations, the quadratic formula and the discriminant; this unit uses those tools. You will need a scientific calculator, graph paper, and if possible a free graphing tool such as Desmos.

Every lesson has worked examples with full solutions, diagrams, a hands-on activity, lots of practice with answers and a short quiz. The mathematical processes (problem solving, reasoning and proving, reflecting, selecting tools and strategies, connecting, representing and communicating) are woven through every lesson.

Curriculum expectations covered: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions and Applications, Grade 11, University/College Preparation (MCF3M), strand A, Quadratic Functions: A2.1 to A2.11 (connecting graphs and equations of quadratic functions) and A3.1 to A3.3 (solving problems involving quadratic functions).

In this unit you will learn the mathematics of things that grow or shrink by the same factor over and over: populations, bacteria, savings, the value of a car, a cooling drink, a bouncing ball and radioactive samples.

You start with powers. You will use the exponent rules for multiplying, dividing and raising a power to a power, evaluate powers with zero and negative exponents and fraction, decimal and negative bases, and discover what rational (fraction) exponents such as 9^(1/2) and 8^(2/3) mean. Then you will graph exponential functions f(x) = aˣ, explain why they are functions, describe their key properties (domain, range, intercepts, increasing or decreasing, asymptote), and tell them apart from linear and quadratic functions using tables, graphs and equations. Finally you will collect data from experiments, build exponential models, recognize growth and decay with their real-world restrictions, and solve problems about populations, interest, half-life, depreciation, bouncing balls and cooling.

Remember from earlier grades: in Grade 9 you learned the exponent laws and in Grade 10 you explained zero and negative exponents and compared y = x² with y = 2ˣ. The Quadratic Functions unit of this course introduced functions, function notation, domain and range. The Financial Mathematics unit that follows uses these ideas for compound interest and annuities. You need a scientific calculator with a power key; a free graphing tool (such as Desmos or GeoGebra) is helpful but every answer can be found by hand or with a scientific calculator.

Writing fraction exponents: on this site "8 to the power 2/3" is written 8^(2/3), exactly as you type it on a calculator. Whole-number exponents are written the usual way (8²).

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions and Applications, Grade 11, University/College Preparation (MCF3M), strand B. Exponential Functions: B1.1 to B1.6 (connecting graphs and equations of exponential functions) and B2.1 to B2.3 (solving problems involving exponential functions). B3 (financial problems) is taught in the Financial Mathematics 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 learn the mathematics behind saving, investing and borrowing. You will start by comparing simple and compound interest with tables and graphs, then use the compound interest formula A = P(1 + i)ⁿ to find future values and P = A(1 + i)⁻ⁿ to find present values: how much to invest now to reach a goal. You will discover, with spreadsheets and graphs, that compound interest is exponential growth, connecting this unit to the unit Exponential Functions. You will solve for the interest rate and the number of compounding periods using rational exponents, tables, graphs, a TVM Solver and spreadsheets. Then you will study annuities, the regular deposits and payments behind savings plans and loans: their future value, present value and regular payment, amortization tables and the total interest on a loan. You will investigate how changing the payment, the payment frequency, the interest rate or the compounding period changes an annuity, and finish by making real saving, investing and borrowing decisions with current Canadian rates and rules.

Remember from earlier grades: in Grade 8 you calculated simple interest and compound interest compounded yearly; in Grade 9 you used A = P(1 + i)ⁿ with different compounding periods, compared loans and built an amortization table in a spreadsheet. Grade 11 goes further: present value, solving for the rate and time, and annuities. You need a scientific calculator; a TVM Solver (on a graphing calculator or a free website) and a spreadsheet are very helpful. Every lesson has worked examples, a hands-on activity, lots of practice with full answers, and a short quiz.

Real numbers: rates in the worked examples are practice values, not offers. Where a lesson uses real Canadian rates or rules (Bank of Canada policy rate, prime rate, TFSA limit, CDIC coverage, credit card grace periods, the criminal interest rate, student loan interest), they are dated (checked October 2026) and sourced. These change often, so check the source before any real decision.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Functions and Applications, Grade 11, University/College Preparation (MCF3M), strand B. Exponential Functions, overall expectation B3 (solving financial problems involving exponential functions): specific expectations B3.1 to B3.7. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson. The sibling unit Exponential Functions covers B1 and B2.

In this unit you will measure distances and heights you cannot reach directly. You will start by sharpening your right-triangle trigonometry: naming the sides of a right triangle, using the primary ratios sine, cosine and tangent to find sides and angles, and applying them to angles of elevation and depression and to bearings in surveying and navigation. Then you will solve problems that need two right triangles that share a side, in two dimensions and then in three.

In the second half of the unit you will leave right triangles behind. You will discover and verify the sine law and the cosine law (by measuring, with a spreadsheet or with dynamic geometry software, and by seeing where each law comes from), learn the conditions that tell you which law to use, and use both laws to solve acute triangles in real-world problems from surveying, navigation and building construction.

Remember from Grade 10: Principles of Mathematics (MPM2D) on this site taught similar triangles, the primary trigonometric ratios, and a first look at the sine law and the cosine law in acute triangles. If you took Foundations of Mathematics (MFM2P) instead, you will have used the primary ratios in right triangles, and the sine and cosine laws may be new; the first lessons here review everything you need. You will need a scientific calculator in degree mode.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Functions and Applications, Grade 11, University/College Preparation (MCF3M), strand C. Trigonometric Functions, overall expectation C1 (solving problems involving trigonometry in acute triangles using the sine law and the cosine law) and its specific expectations C1.1 to C1.5. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson. The other overall expectations of this strand (C2 and C3) are taught in the next unit, Sine Functions.

This unit is about things that repeat: tides, daylight, Ferris wheels, swimmers doing lengths and the electricity in your home. You will start by describing periodic functions (cycle, period, axis and amplitude) and using the period to make predictions. Next you will extend the sine ratio from right triangles to every angle from 0° to 360° on a circle, and use it to build the sine function, f(x) = sin x, and learn its key properties. Then you will transform the sine graph with a (vertical stretches, compressions and reflections), d (horizontal translations) and c (vertical translations), and connect these to changes in real situations. Finally you will decide when a situation is sinusoidal, collect and graph your own data, build models of the form y = a sin(x − d) + c, and pose and solve problems with them.

This unit follows Trigonometry of Acute Triangles, which reviews the primary trigonometric ratios and the sine and cosine laws. Sine functions return in college courses such as MCT4C, and in MCR3U and MHF4U for students who take those courses.