Ontario Grade 12 MHF4U Advanced Functions (Grade 12, University Preparation)
Chapters: 4
1. A. Exponential and Logarithmic Functions
1 Evaluating Logarithmic Expressions · 2 Connecting Graphs and Equations of Logarithmic Functions · 3 Solving Exponential and Logarithmic Equations
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
2. B. Trigonometric Functions
1 Understanding and Applying Radian Measure · 2 Connecting Graphs and Equations of Trigonometric Functions · 3 Solving Trigonometric Equations
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
3. C. Polynomial and Rational Functions
1 Connecting Graphs and Equations of Polynomial Functions · 2 Connecting Graphs and Equations of Rational Functions · 3 Solving Polynomial and Rational Equations · 4 Solving Inequalities
- Polynomial Functions and Their Graphs – A polynomial function is y = aₙxⁿ + … + a₁x + a₀ with whole-number powers. The degree n and the leading coefficient aₙ fix the end behaviour. Each real zero r gives a factor (x − r); the graph crosses the x-axis at a zero of odd multiplicity and touches (bounces) at a zero of even multiplicity. A degree-n polynomial has at most n real zeros and at most n − 1 turning points, and exactly n zeros when complex ones are counted. The average rate of change between two points is the slope of the secant line.
- Rational Functions: Graphs, Asymptotes and Sketching – A rational function is one polynomial divided by another, like y = (2x + 1)/(x − 1). It is not defined where the bottom is zero. Near those x-values the graph shoots up or down along a vertical asymptote. Far out, it settles near a horizontal (or slant) asymptote. Find domain, asymptotes, intercepts and holes, then sketch.
- Solving Polynomial and Rational Equations – To solve a polynomial equation p(x) = 0, find where the graph of y = p(x) meets the x-axis. For a cubic or quartic: guess one root from the divisors of the constant term, divide it out, and solve what is left. Biquadratics like x⁴ − 5x² + 4 = 0 become quadratics with t = x². Rational equations P(x)/Q(x) = 0 need P(x) = 0 and Q(x) ≠ 0. Inequalities use a sign chart.
4. D. Characteristics of Functions
1 Understanding Rates of Change · 2 Combining Functions · 3 Using Function Models to Solve Problems
- Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
- Relations and Functions: Cartesian Product, Domain, Range, Graphs – An ordered pair (a, b) has a first and a second place, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and n(A × B) = n(A) × n(B). A relation from A to B is any subset of A × B; its domain is the set of first elements and its range the set of second elements, while B is the co-domain. A function is a special relation where every element of A has exactly one image. We study constant, identity, polynomial, rational, modulus, signum, exponential, log and greatest integer functions with their graphs, and add, subtract, multiply and divide functions point by point.
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.