Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Solve quadratic equations using various methods.
• Sketch graphs of quadratic functions (parabolas).
• Apply the sine, cosine, and area rules to solve triangles.

Lesson summary

This week is dedicated to comprehensive revision and examination preparation. We will be focusing on key areas from the previous weeks to solidify your understanding and build confidence for upcoming assessments. Mathematics is not just a subject confined to the classroom; it is a crucial tool for navigating everyday life, from budgeting your finances to understanding data and making informed decisions. In the South African context, a strong foundation in mathematics is essential for future careers in fields like engineering, finance, technology, and even entrepreneurship. It empowers you to analyze social and economic challenges and contribute meaningfully to society.

Lesson notes

2.1 Quadratic Equations A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠

0. Methods for solving quadratic equations: Factorization: This involves expressing the quadratic expression as a product of two linear factors.

Example: Solve x² - 5x + 6 = 0 (x - 2)(x - 3) = 0 Therefore, x = 2 or x = 3 Completing the Square: This method involves manipulating the equation to create a perfect square trinomial.

Example: Solve x² + 4x - 5 = 0 x² + 4x = 5 x² + 4x + (4/2)² = 5 + (4/2)² x² + 4x + 4 = 9 (x + 2)² = 9 x + 2 = ±3 x = 1 or x = -5 Quadratic Formula: This is a general formula that can be used to solve any quadratic equation.

The formula is: x = (-b ± √(b² - 4ac)) / 2a

Example: Solve 2x² + 3x - 2 = 0 a = 2, b = 3, c = -2 x = (-3 ± √(3² - 4(2)(-2))) / (2(2)) x = (-3 ± √25) / 4 x = (-3 ± 5) / 4 x = 1/2 or x = -2 Why These Methods Work: Factorization relies on the zero product property (if ab = 0, then a = 0 or b = 0). Completing the square transforms the equation into a form where the square root property can be applied. The quadratic formula is derived by completing the square on the general quadratic equation.

Example Application: A farmer in KwaZulu-Natal wants to fence a rectangular field. He has 100 meters of fencing. What are the dimensions of the field that will maximize the area? Let length = l, width = w. 2l + 2w = 100 => l + w = 50 => l = 50 - w. Area A = l w = (50 - w) w = 50w - w². To maximize, take derivative and set to zero (Calculus is allowed here as an example of application): dA/dw = 50 - 2w = 0 => w =

2

5. Therefore, l = 50 - 25 =

2

5. The dimensions are 25m x 25m, maximizing area. 2.2 Quadratic Functions (Parabolas) A quadratic function is a function of the form f(x) = ax² + bx + c, where a ≠

0. The graph of a quadratic function is a parabola.

Key features of a parabola: Turning Point (Vertex): The maximum or minimum point of the parabola. The x-coordinate of the turning point is given by x = -b / 2a.

Axis of Symmetry: A vertical line that passes through the turning point. Its equation is x = -b / 2a.

Y-intercept: The point where the parabola intersects the y-axis. It is found by setting x = 0 in the equation.

X-intercept(s): The point(s) where the parabola intersects the x-axis. They are found by setting f(x) = 0 and solving the resulting quadratic equation.

Sketching a parabola: Find the turning point. Find the y-intercept. Find the x-intercept(s) (if they exist). Determine the shape of the parabola (if a > 0, the parabola opens upwards; if a (x - 3)(x + 1) =

0. X-intercepts: (3, 0) and (-1, 0) Since a = 1 > 0, the parabola opens upwards. Sketch the parabola using the information above. 2.3 Sine, Cosine, and Area Rules These rules are used to solve triangles that are not right-angled triangles.

Sine Rule: a/sinA = b/sinB = c/sinC Cosine Rule: a² = b² + c² - 2bc cosA (or variations to find cosA: cosA = (b² + c² - a²) / 2bc)

Area Rule: Area = (1/2)ab sinC Choosing the correct rule: Sine Rule: Use when you have two angles and a side opposite one of them, or two sides and an angle opposite one of them.

Cosine Rule: Use when you have three sides, or two sides and the included angle.

Area Rule: Use when you have two sides and the included angle.

Example: In triangle ABC, AB = 5 cm, BC = 7 cm, and angle ABC = 60°. Find the length of A

C. Using the cosine rule: AC² = AB² + BC² - 2(AB)(BC)cos(ABC) AC² = 5² + 7² - 2(5)(7)cos(60°) AC² = 25 + 49 - 70(1/2) AC² = 39 AC = √39 cm Example Application: Imagine surveying a piece of land for a new housing development in Gauteng. The Sine and Cosine rules would be used to calculate distances and angles that are difficult to measure directly due to terrain. The Area rule helps calculate the land area for planning purposes. 2.4 Arithmetic and Geometric Sequences and Series Arithmetic Sequence: A sequence where the difference between consecutive terms is constant (common difference, d). The nth term is given by Tn = a + (n - 1)d. The sum of the first n terms is given by Sn = (n/2)[2a + (n - 1)d] or Sn = (n/2)(a + l), where l is the last term.

Geometric Sequence: A sequence where the ratio between consecutive terms is constant (common ratio, r). The nth term is given by Tn = ar^(n-1). The sum of the first n terms is given by Sn = a(r^n - 1) / (r - 1), where r ≠

1. The sum to infinity (when |r| (x - 2)² + (y + 3)² =

1

6. Finding the centre and radius from the equation: If you have the equation of a circle, you can identify the centre and radius by comparing it to the standard form.

Example: Given the equation (x + 1)² + (y - 5)² = 9, find the centre and radius. The centre is (-1, 5) and the radius is √9 =

3. Example Application: In urban planning in cities like Johannesburg, knowing the equation of a circle can be useful for designing roundabouts or planning infrastructure around circular features. Guided Practice (With Solutions)

Question 1: Solve the quadratic equation 3x² - 7x + 2 = 0 using factorization.