Advanced Functions, Grade 12 (University)

Unit 1: Exponential and Logarithmic Functions

Evaluating and graphing logarithmic functions as the inverse of exponential functions, applying the laws of logarithms, and solving exponential and logarithmic equations, including real-world applications.

• The logarithm as the inverse of exponentiation
• Approximating logarithms with technology
• Connecting logarithmic and exponential equations

• Deriving and verifying the laws of logarithms
• Applying the laws of logarithms to simplify and evaluate

• Key features of the graph of f(x) = log x
• Logarithmic and exponential functions as inverses
• The roles of parameters in transformed logarithmic functions
• Solving real-world exponential and logarithmic problems from a graph
• Simplifying equivalent logarithmic and exponential expressions

• Solving exponential equations by finding a common base
• Solving exponential equations using logarithms
• Solving simple logarithmic equations algebraically
• Solving real-world problems with exponential and logarithmic equations

Unit 2: Trigonometric Functions

Understanding and applying radian measure, connecting graphs and equations of trigonometric functions and their reciprocals, and solving trigonometric equations and proving identities.

• The radian as an alternative angle-measure unit
• Representing radian measure in terms of π and as a rational number
• Trigonometric ratios of angles in radians, with technology
• Exact trigonometric values for special angles in radians

• Sketching sine and cosine graphs in radians
• Connecting the tangent ratio to the tangent function
• Graphing and describing the reciprocal trigonometric functions
• Amplitude, period, and phase shift of sinusoidal functions
• Sketching transformed sine and cosine graphs
• Writing the equation of a sinusoidal function from its graph or properties
• Solving real-world problems with trigonometric functions in radians

• Recognizing equivalent trigonometric expressions
• Compound angle formulas and exact trig values
• Understanding trigonometric identities and disproving non-identities
• Proving trigonometric identities
• Solving linear and quadratic trigonometric equations

Unit 3: Polynomial and Rational Functions

Connecting graphs and equations of polynomial and rational functions, factoring and solving polynomial and rational equations and inequalities, using the remainder and factor theorems.

• Recognizing polynomial expressions and functions
• Comparing numeric, graphical, and algebraic polynomial representations
• Key features of polynomial function graphs
• Distinguishing polynomial functions from sinusoidal and exponential functions
• Connecting factored form to x-intercepts and sketching the graph
• Roles of a, k, d, and c as transformations of general functions
• Determining a polynomial equation from given conditions
• The family of polynomial functions with given zeros
• Properties of even and odd polynomial functions

• Key features of rational functions that are reciprocals of linear/quadratic functions
• Key features of rational functions with linear numerator and denominator
• Sketching a simple rational function from its key features

• Verifying the remainder theorem and the factor theorem
• Factoring polynomial expressions of degree up to four
• Connecting real roots of a polynomial equation to x-intercepts
• Solving polynomial equations of degree up to four
• Connecting real roots of a rational equation to x-intercepts
• Solving simple rational equations algebraically
• Solving real-world problems with polynomial and rational functions

• The difference between solving an equation and an inequality
• Solving polynomial and rational inequalities by graphing
• Solving linear and factorable polynomial inequalities algebraically

Unit 4: Characteristics of Functions

Understanding average and instantaneous rate of change, combining functions, and comparing and modelling with functions — the calculus on-ramp of the course.

• Real-world rates of change and their representations
• Zero, constant, and changing rates of change
• Sketching a graph from a rate-of-change description
• Calculating and interpreting average rates of change
• Connecting instantaneous and average rates of change
• Approximating instantaneous rate of change from average rates
• Connecting secant and tangent slopes to rate of change
• Approximating the slope of a tangent using secants
• Solving average and instantaneous rate-of-change problems

• Key features of functions formed by combining functions
• Real-world applications of combinations of functions
• Even, odd, and increasing/decreasing properties of combined functions
• Composing two functions numerically and graphically
• Composing two functions algebraically and stating domain and range
• Solving real-world problems with the composition of two functions
• The composition of a function and its inverse maps a number onto itself
• Connecting transformations to composition with a linear function

• Comparing the characteristics of different function families
• Solving equations and inequalities not accessible algebraically
• Solving real-world problems by reasoning with function models