Ontario Grade 11 MCF3M Functions and Applications (Grade 11, University/College Preparation)
Chapters: 3
1. A. Quadratic Functions
1 Solving Quadratic Equations · 2 Connecting Graphs and Equations of Quadratic Functions · 3 Solving Problems Involving Quadratic Functions
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
- Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
2. B. Exponential Functions
1 Connecting Graphs and Equations of Exponential Functions · 2 Solving Problems Involving Exponential Functions · 3 Solving Financial Problems Involving Exponential Functions
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
3. C. Trigonometric Functions
1 Applying the Sine Law and the Cosine Law in Acute Triangles · 2 Connecting Graphs and Equations of Sine Functions · 3 Solving Problems Involving Sine Functions
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
- 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.