Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Solve quadratic equations using various methods.
• Sketch and analyze graphs of functions.
• Apply trigonometric identities and solve trig equations.
• Calculate probabilities of simple and compound events.
• Work with financial mathematics concepts.

Lesson summary

This week is dedicated to consolidating our understanding and enhancing our problem-solving skills across several key areas in Grade 11 Mathematics. Strong exam preparation is crucial not only for achieving good marks but also for building a solid foundation for further studies in mathematics and related fields. Mastering these concepts will empower you to tackle real-world problems and make informed decisions in your daily lives. For example, understanding functions can help you analyze trends in stock markets (crucial in a country like South Africa with significant economic inequality), while trigonometry can be used in surveying and construction, essential for infrastructure development.

Lesson notes

This week focuses on integrating knowledge from previous topics, which includes quadratic equations, functions and their graphs, trigonometry, probability and financial mathematics. Each of these areas have significant real-world applications. a)

Quadratic Equations: A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠

0. Solving quadratic equations is fundamental to many mathematical problems and is also used widely in physics (e.g., projectile motion) and engineering.

Methods of Solving: Factorization: This involves expressing the quadratic expression as a product of two linear factors. This method is effective when the roots are rational.

Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It is generally used when factorization is not straightforward.

Quadratic Formula: The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a general solution for any quadratic equation.

The Discriminant: The discriminant (Δ = b² - 4ac) determines the nature of the roots: Δ > 0: Two distinct real roots Δ = 0: One real root (repeated root) Δ x = 1/3 or x = -1 b)

Functions and Their Graphs: A function is a relationship between two sets of numbers (input and output) where each input value corresponds to exactly one output value. We will focus on linear, quadratic, exponential, and hyperbolic functions.

Linear Function: y = mx + c, where m is the gradient and c is the y-intercept.

Quadratic Function: y = ax² + bx + c, whose graph is a parabola. Key features include the vertex (turning point), axis of symmetry, intercepts, and shape.

Exponential Function: y = a x , where a > 0 and a ≠

1. These functions show exponential growth or decay.

Hyperbolic Function: y = k/x, where k is a constant. Key features include asymptotes at x = 0 and y =

0. Example 4: Sketch the graph of y = x² - 4x +

3. Solution: Find the x-intercepts: Solve x² - 4x + 3 = 0. (x - 1)(x - 3) = 0 => x = 1, x = 3 Find the y-intercept: Set x = 0 => y = 3 Find the vertex: Complete the square: y = (x - 2)² -

1. Vertex is (2, -1). Sketch the parabola using these key points. c)

Trigonometry: Trigonometry deals with the relationships between angles and sides of triangles. It's used extensively in navigation, surveying, and engineering.

Trigonometric Ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Trigonometric Identities: These are equations that are true for all values of the variables.

Important identities include: sin² θ + cos² θ = 1 tan θ = sin θ / cos θ sin (90° - θ) = cos θ cos (90° - θ) = sin θ Example 5: Simplify the expression (sin² θ + cos² θ) / cos θ.

Solution: Use the identity sin² θ + cos² θ =

1. The expression becomes 1 / cos θ = sec θ.

Example 6: Solve the equation 2sin θ = 1 for 0° ≤ θ ≤ 360°.

Solution: Divide by 2: sin θ = 1/2 Find the reference angle: θ = sin⁻¹(1/2) = 30° Since sin θ is positive in the first and second quadrants, the solutions are: θ = 30° and θ = 180° - 30° = 150°. d)

Probability: Probability is the measure of the likelihood that an event will occur. It is essential for understanding risk and making informed decisions.

Basic Probability: P(event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Compound Events: Events involving multiple steps or outcomes.

Independent Events: The outcome of one event does not affect the outcome of the other. P(A and B) = P(A)

P(B)

Dependent Events: The outcome of one event affects the outcome of the other. P(A and B) = P(A) P(B|A)

Tree Diagrams and Venn Diagrams: Useful for visualizing and calculating probabilities of compound events.

Example 7: A bag contains 3 red balls and 5 blue balls. What is the probability of drawing a red ball, then (without replacement) another red ball?

Solution: P(Red on first draw) = 3/8 P(Red on second draw | Red on first draw) = 2/7 (since one red ball has been removed) P(Red then Red) = (3/8) * (2/7) = 6/56 = 3/28 e)

Financial Mathematics: Financial mathematics deals with the time value of money and is crucial for making informed financial decisions.

Simple Interest: A = P(1 + nr), where A is the accumulated amount, P is the principal, n is the number of years, and r is the interest rate.

Compound Interest: A = P(1 + i) n , where i is the interest rate per period and n is the number of periods.

Depreciation: The decrease in the value of an asset over time.

Straight-line Depreciation: A = P(1 - nr)

Reducing Balance Depreciation: A = P(1 - i) n Example 8: You invest R5000 at an interest rate of 8% per annum compounded annually. How much will you have after 5 years?