Ontario Grade 12 MCT4C Mathematics for College Technology (Grade 12, College Preparation)
Chapters: 4
1. A. Exponential Functions
1 Solving Exponential Equations Graphically · 2 Solving Exponential Equations Algebraically
- 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.
- 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. Polynomial Functions
1 Investigating Graphs of Polynomial Functions · 2 Connecting Graphs and Equations of Polynomial Functions · 3 Solving Problems Involving Polynomial Equations
- 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.
3. C. Trigonometric Functions
1 Applying Trigonometric Ratios · 2 Connecting Graphs and Equations of Sinusoidal Functions · 3 Solving Problems Involving Sinusoidal 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.
4. D. Applications of Geometry
1 Modelling With Vectors · 2 Solving Problems Involving Geometry · 3 Solving Problems Involving Circle Properties
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
- Circle Theorems – Circle theorems are a small set of angle rules that are always true inside a circle. The angle at the centre is twice the angle at the edge. The angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. A tangent meets the radius at 90°, two tangents from one point are equal, and the tangent–chord angle equals the angle in the alternate segment.