MCR3U Grade 11 Functions — Exam Formula Sheet & Review
Ontario Ministry of Education Curriculum Aligned
1. Function Notation & Transformations
Standard Transformation Form: y = a · f(k(x - d)) + c
- |a| > 1: Vertical stretch by factor of |a|; 0 < |a| < 1: Vertical compression by factor of |a|. If a < 0, reflection in the x-axis.
- |k| > 1: Horizontal compression by factor of 1/|k|; 0 < |k| < 1: Horizontal stretch by factor of 1/|k|. If k < 0, reflection in the y-axis.
- d: Horizontal translation. If d > 0, shift right d units. If d < 0, shift left |d| units.
- c: Vertical translation. If c > 0, shift up c units. If c < 0, shift down |c| units.
- Coordinate Mapping Rule:
(x, y) → (x/k + d, a·y + c) - Inverse Function f⁻¹(x): Swap x and y, then isolate y. Reflects graph over the line y = x.
2. Quadratics & Radicals
Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a)
Discriminant D = b² - 4ac:
- D > 0: Two distinct real roots (two x-intercepts).
- D = 0: Exactly one real root (parabola vertex rests on the x-axis).
- D < 0: Zero real roots (parabola does not cross the x-axis).
Completing the Square: y = ax² + bx + c → a[x + b/(2a)]² + [c - b²/(4a)] with vertex at (h, k).
3. Trigonometry & Exact Special Angles
Primary Trig Ratios: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
Reciprocal Trig Ratios: csc θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ
Pythagorean & Quotient Identities: sin² θ + cos² θ = 1, tan θ = sin θ / cos θ
| Angle (θ) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° (π/6) | 1/2 | √3 / 2 | 1 / √3 = √3 / 3 |
| 45° (π/4) | 1 / √2 = √2 / 2 | 1 / √2 = √2 / 2 | 1 |
| 60° (π/3) | √3 / 2 | 1/2 | √3 |
4. Sequences & Series
Arithmetic Sequence: General Term: t_n = a + (n - 1)d; Sum: S_n = (n/2)[2a + (n - 1)d] = (n/2)(a + t_n)
Geometric Sequence: General Term: t_n = a · r^(n - 1); Sum: S_n = [a(r^n - 1)] / (r - 1) (for r ≠ 1)
5. Exponential Functions & Financial Math
Compound Interest: A = P(1 + i)^n (where i = annual rate / compounding periods, n = total periods)
Ordinary Simple Annuity (Future Value): FV = R · [(1 + i)^n - 1] / i
Present Value Annuity: PV = R · [1 - (1 + i)^(-n)] / i