Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Analyze and Sketch Functions (Linear, Quadratic, Hyperbolic, Exponential, Logarithmic)
• Apply Trigonometric Identities to solve equations and problems
• Solve complex Euclidean Geometry problems with justification
• Integrate concepts from different sections of the syllabus
• Practice effective exam time management

Lesson summary

This week marks a critical phase in our Grade 12 Mathematics journey: focused revision and preparation for the final examinations. We will consolidate our understanding of essential concepts, refine problem-solving techniques, and build confidence in tackling a wide range of question types. Mathematics is fundamental not only for further studies in fields like engineering, finance, and science, but also for everyday problem-solving, critical thinking, and informed decision-making. In a South African context, mathematical literacy is crucial for understanding economic trends, participating in democratic processes, and navigating a rapidly changing technological landscape.

Lesson notes

2.1 Functions Linear Functions: In the form f(x) = mx + c, where m is the gradient and c is the y-intercept. Crucial for modeling linear relationships, like taxi fares based on distance.

Quadratic Functions: In the form f(x) = ax² + bx + c or f(x) = a(x-p)² + q. The latter form reveals the turning point (p, q). Essential for modeling projectile motion and optimization problems.

Hyperbolic Functions: In the form f(x) = a/(x + p) + q. Have vertical asymptote x = -p and horizontal asymptote y = q. Useful in understanding inverse relationships, such as the relationship between price and demand in economics.

Exponential Functions: In the form f(x) = a^x or f(x) = a^(x + p) + q. Show exponential growth or decay. Horizontal asymptote y = q. Essential in modelling population growth, compound interest and the spread of disease.

Logarithmic Functions: In the form f(x) = log_a(x) or f(x) = log_a(x + p) + q, where a > 0 and a ≠

1. The inverse of exponential functions. Vertical asymptote x = -p. Used in measuring sound intensity (decibels) and earthquake magnitude (Richter scale).

Key features to analyze: Intercepts: Where the graph crosses the x-axis (x-intercepts, found by setting y = 0) and y-axis (y-intercept, found by setting x = 0).

Asymptotes: Lines that the graph approaches but never touches (vertical and horizontal).

Turning Points: Maximum or minimum points on the graph (for quadratic functions, the vertex; for other functions, found using calculus if required).

Axes of Symmetry: A line that divides the graph into two symmetrical halves (for quadratic functions, the vertical line through the turning point).

Domain and Range: The set of all possible input values (x) and output values (y), respectively.

Transformations: Understanding how p and q in f(x + p) + q affect the graph. p causes a horizontal shift (left if positive, right if negative) and q causes a vertical shift (up if positive, down if negative). A negative sign in front of the function or x value results in reflection across the x or y axis respectively.

Worked example

Sketch the graph of f(x) = -2/(x - 1) +

3. State the domain, range, and asymptotes.

Solution:

Asymptotes: Vertical asymptote: x = 1; Horizontal asymptote: y =

3. Shape: Hyperbola. The negative sign indicates a reflection in the x-axis of a standard hyperbola.

Intercepts: To find the y-intercept, set x = 0: f(0) = -2/(0 - 1) + 3 =

5. The y-intercept is (0, 5). To find the x-intercept, set f(x) = 0: 0 = -2/(x-1) + 3. 2/(x-1) = 3. 2 = 3(x-1). 2 = 3x - 3. 5 = 3x. x = 5/

3. The x-intercept is (5/3, 0).

Sketch: Plot the asymptotes and intercepts, and draw the hyperbola.

Domain: All real numbers except x = 1 or x ∈ ℝ, x ≠

1. Range: All real numbers except y = 3 or y ∈ ℝ, y ≠ 3.

2.2 Trigonometry

Trigonometric Ratios: Sine, cosine, and tangent (SOH CAH TOA). Understand these in the context of right-angled triangles.

Trigonometric Identities: Fundamental identities like sin²θ + cos²θ = 1, tanθ = sinθ/cosθ.

Compound angle formulae: sin(A ± B) = sinAcosB ± cosAsinB, cos(A ± B) = cosAcosB ∓ sinAsin

B. Double angle formulae: sin2A = 2sinAcosA, cos2A = cos²A - sin²A = 1 - 2sin²A = 2cos²A -

1. These are crucial for simplifying expressions and solving equations.