Lesson Notes By Weeks and Term v4 - SHS 3
PATTERNS AND RELATIONS
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PATTERNS AND RELATIONS
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Subject: Mathematics
Class: SHS 3
Term: 1st Term
Week: 12
Grade code: 3.2.2.LI.2
Strand code: 2
Sub-strand code: 2
Content standard code: 3.2.2.CS.1
Indicator code: 3.2.2.LI.2
Theme: ALGEBRAIC REASONING
Subtheme: PATTERNS AND RELATIONS
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This lesson explores the visual representation of quadratic functions. We move beyond simply solving equations algebraically to understanding what they look like and what their shapes tell us. This is crucial because many real-world phenomena, from the path of a thrown stone to the profit a market woman makes, can be modelled by these U-shaped curves called parabolas. By learning to draw and interpret these graphs, we gain the power to find optimal values—like the maximum height of an object or the maximum profit—which is a valuable skill in science, business, and engineering.
Teacher's Note: Begin with a "Talk for Learning" activity. Ask students: "How do we draw the graph of a linear equation like `y = 2x + 1`?" Guide a whole-class discussion to review creating a table of values, plotting points, and drawing a straight line. This activates prior knowledge before introducing curves. A. What is a Quadratic Function?
A quadratic function is a function that can be written in the standard form: `y = ax² + bx + c` where `a`, `b`, and `c` are constants, and `a` is not equal to zero (`a ≠ 0`). If `a` were zero, it would be a linear function!
The graph of a quadratic function is a smooth, U-shaped curve called a parabola. B. Key Features of a Parabola Shape and Orientation: The coefficient `a` (the number multiplying `x²`) tells us which way the parabola opens. If `a > 0` (positive), the parabola opens upwards (like a smile 😊). It has a minimum point. If `a < 0` (negative), the parabola opens downwards (like a frown ☹️). It has a maximum point. Vertex: This is the turning point of the parabola. It is the lowest point on the graph if the parabola opens upwards (a minimum point). It is the highest point on the graph if the parabola opens downwards (a maximum point). The maximum/minimum value of the function is the y-coordinate of the vertex. Axis of Symmetry: This is a vertical line that passes through the vertex and divides the parabola into two perfect mirror images. The equation of this line is `x = k`, where `k` is the x-coordinate of the vertex. y-intercept: This is the point where the graph crosses the y-axis. At this point, `x = 0`. For the function `y = ax² + bx + c`, the y-intercept is always the point (0, c). x-intercepts (Roots or Zeros): These are the points where the graph crosses the x-axis. At these points, `y = 0`. The x-coordinates of these points are the solutions or roots of the quadratic equation `ax² + bx + c = 0`. A parabola can have two x-intercepts, one x-intercept (if the vertex is on the x-axis), or no x-intercepts at all. C. How to Draw the Graph of a Quadratic Function
Step 1: Create a Table of Values. Given a function like `y = x² + 2x - 3` and an interval, say `-4 ≤ x ≤ 2`, choose integer values of `x` in that range and calculate the corresponding `y` values. It is best to do this systematically in a table.