Functions, Grade 11 (University)

Unit 1: Characteristics of Functions

What a function is and how it differs from a relation; function notation, domain, range, and inverses; transformations of functions; quadratic functions as an application; and simplifying polynomial, radical, and rational expressions.

• What makes a relation a function
• Using function notation and evaluating functions
• Explaining domain and range
• Connecting a function's inverse to reverse processes
• Finding a function's inverse numerically and graphically
• Domain and range of an inverse relation
• Finding a function's inverse algebraically

• The roles of a, k, d, and c in y = af(k(x − d)) + c
• Sketching transformed functions

• Finding the number of zeros of a quadratic function
• Finding the maximum or minimum of a quadratic function algebraically
• Solving real-world quadratic function problems
• The family of quadratic functions sharing the same zeros
• Solving the intersection of a linear and a quadratic function

• Simplifying polynomial expressions
• Simplifying radicals and radical expressions
• Simplifying rational expressions and stating restrictions
• Checking whether two algebraic expressions are equivalent

Unit 2: Exponential Functions

Rational exponents and simplifying exponential expressions; key properties, graphs, and transformations of exponential functions; and solving real problems modelled with exponential growth and decay.

• Graphing an exponential relation and defining it as a function
• Finding the value of a power with a rational exponent
• Simplifying and evaluating expressions with integer and rational exponents

• Key properties of exponential functions
• Distinguishing exponential from linear and quadratic functions
• The roles of a, k, d, and c in exponential transformations
• Sketching transformed exponential functions
• Expressing an exponential function using a different base
• Writing an exponential function's equation from its graph or properties

• Collecting and graphing exponential data
• Identifying real-world exponential growth and decay
• Solving real-world exponential function problems

Unit 3: Discrete Functions

Sequences as discrete functions, Pascal's triangle and binomial expansion, arithmetic and geometric sequences and series, and financial applications — simple interest, compound interest, and ordinary simple annuities — as applications of sequences and series.

• Connecting sequences to discrete functions
• Describing a recursive procedure for a sequence
• Connecting the nth-term formula to function notation
• Representing a sequence algebraically in multiple ways
• Recursive patterns in the Fibonacci sequence and Pascal's triangle
• Pascal's triangle and expanding binomials

• Identifying a sequence as arithmetic, geometric, or neither
• Finding the general-term formula for arithmetic and geometric sequences
• Finding the sum formula for arithmetic and geometric series
• Solving problems with arithmetic and geometric sequences and series

• Connecting simple interest to arithmetic sequences and linear growth
• Connecting compound interest to geometric sequences and exponential growth
• Solving for amount, principal, or interest rate with the compound interest formula
• Solving for the number of compounding periods
• What an annuity is, with real examples
• Connecting annuities to geometric series and exponential growth
• Effects of changing an annuity's conditions
• Solving for the amount, present value, or payment of an annuity

Unit 4: Trigonometric Functions

Trigonometric ratios for angles up to 360°, reciprocal ratios and identities, solving triangles with the sine law and cosine law, and the sine and cosine functions — their graphs, transformations, and use modelling periodic real-world phenomena.

• Exact trig values for the special angles
• Finding trig values for angles from 0° to 360°
• Finding two angles with the same trig ratio value
• Defining the secant, cosecant, and cotangent ratios
• Proving simple trigonometric identities
• Solving right and oblique triangle problems in two dimensions
• Solving right and oblique triangle problems in three dimensions

• Key properties of periodic functions
• Predicting future behaviour of a periodic relationship
• Connecting the sine and cosine ratios to the sine and cosine functions
• Sketching sine and cosine graphs and their key properties
• The roles of a, k, d, and c in sinusoidal transformations
• Finding amplitude, period, phase shift, domain, and range from an equation
• Sketching transformed sine and cosine graphs
• Writing a sinusoidal function's equation from its graph or properties

• Collecting and graphing sinusoidal data
• Identifying real-world periodic and sinusoidal functions
• Using sinusoidal functions to model phenomena without angles
• Predicting the effect of varying conditions on a periodic model
• Solving real-world sinusoidal function problems