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Chapter 2: Quadratic Functions

Form 4 Additional Mathematics Bab 2: Quadratic Functions

2.1 Quadratic Equations and Inequalities

  • General form of a quadratic equation: ax2 + bx + c = 0, where a, b, c are constants and a ≠ 0. (p.36)
  • Roots of ax2 + bx + c = 0 are the x-coordinates of the intersection points of y = ax2 + bx + c with the x-axis. (p.37)
  • Completing the square: move the constant to the right, make the coefficient of x2 equal to 1, add (coefficient of x / 2)2 to both sides. (p.37)
  • Quadratic formula: x = (−b ± √(b2 − 4ac)) / 2a. (p.38)
  • Sum of roots: α + β = −b/a. Product of roots: αβ = c/a. (p.39)
  • Equation from roots: x2 − (sum of roots)x + (product of roots) = 0. (p.39)
  • Useful identity: α2 + β2 = (α + β)2 − 2αβ. (p.40)
  • Quadratic inequalities can be solved by graph sketching, number line or table methods. (p.41)
  • For (x − a)(x − b) with a < b: if > 0 then x < a or x > b; if < 0 then a < x < b. (p.42)

2.2 Types of Roots of Quadratic Equations

  • Discriminant D = b2 − 4ac. D > 0: two different real roots. D = 0: two equal real roots. D < 0: no real roots. (p.45)
  • The square root of a negative number is an imaginary root, represented using i = √(−1). (p.45)
  • Arrange the equation in general form before finding the discriminant; use D = 0, D > 0 or D < 0 to find values or ranges of a constant. (p.47)

2.3 Quadratic Functions

  • General form: f(x) = ax2 + bx + c (a ≠ 0). The graph is a parabola symmetrical about an axis through its minimum or maximum point. (p.49)
  • Effect of a: affects shape and width (a > 0 minimum point, a < 0 maximum point; larger |a| gives a narrower graph). The y-intercept is unchanged. (p.50)
  • Effect of b: only changes the position of the vertex relative to the y-axis. Effect of c: moves the graph vertically up or down. (p.50)
  • Position of graph vs discriminant: D > 0 intersects the x-axis at two points; D = 0 touches the x-axis at one point; D < 0 does not intersect the x-axis. (p.52)
  • Vertex form: f(x) = a(x − h)2 + k, vertex (h, k), axis of symmetry x = h. If a > 0, k is the minimum value; if a < 0, k is the maximum value. (p.54)
  • General form has vertex at (−b/2a, f(−b/2a)) and axis x = −b/2a. (p.54)
  • Intercept form: f(x) = a(x − p)(x − q), where p and q are the x-intercepts; axis of symmetry x = (p + q)/2. (p.54)
  • Vertex form to general form: expansion. General to vertex form: completing the square. General to intercept form: factorisation or formula. (p.55)
  • In f(x) = a(x − h)2 + k: changing h shifts the graph horizontally; changing k shifts it vertically; changing a changes the width. (p.58)
  • Sketching steps: determine shape from a, find D for position, find vertex, find x-intercepts (f(x) = 0), find y-intercept f(0), then draw a smooth parabola. (p.60)
  • Maximum/minimum problems: express in vertex form or use t = −b/2a; time to reach the ground is found by solving h(t) = 0. (p.62)
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