Pure: Quadratic Functions

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Grade 1 · Term 1

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About A/L Combined Mathematics: Combined Mathematics consists of Pure Mathematics and Applied Mathematics across two years.

• Identify coefficients: For f(x) = -x² + 2x + 3, we have a = -1, b = 2, c = 3. 2. Determine the direction of opening: Since a = -1 (which is < 0), the parabola opens downwards, indicating a maximum point. 3. Find the axis of symmetry: The formula is x = -b/(2a). So, x = -2/(2*(-1)) = -2/(-2) = 1. The axis of symmetry is x = 1. 4. Find the vertex: Substitute x = 1 into the function: f(1) = -(1)² + 2(1) + 3 = -1 + 2 + 3 = 4. The vertex is (1, 4). This is the maximum point. 5. Find the x-intercepts (zeros): Set f(x) = 0: -x² + 2x + 3 = 0. Multiply by -1: x² - 2x - 3 = 0. Factor the quadratic: (x - 3)(x + 1) = 0. So, x = 3 or x = -1. The x-intercepts are (3, 0) and (-1, 0). 6. Find the y-intercept: Set x = 0: f(0) = -(0)² + 2(0) + 3 = 3. The y-intercept is (0, 3). 7. Sketch the graph: Plot the vertex (1, 4), x-intercepts (3, 0) and (-1, 0), and y-intercept (0, 3). Draw a smooth parabola opening downwards, symmetrical about the line x = 1, passing through these points. ⚠ Tricky: Remembering to handle the negative sign of 'a' correctly when calculating the axis of symmetry and vertex. ⚠ Tricky: Factoring quadratics with a negative leading coefficient; it's often easier to multiply the entire equation by -1 first. ✗ Common mistake: Incorrectly calculating -b/(2a), especially with negative 'b' or 'a' values
• For (a) 2x² - 3x + 1 = 0:
• Identify coefficients for the original equation: For 2x² - 5x + 1 = 0, a=2, b=-5, c=1. 2. Find the sum and product of the original roots (α and β):
• Important Points to Remember

Source: Idasara knowledge pack — english/AL_SCIENCE/combined_mathematics v13, short_notes: Lesson 15 - Competency 3 _ Analyses Quadratic Functions.

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