Chapter 2: Quadratic Functions

2.1 Quadratic Equations and Inequalities

1. Solving Quadratic Equations

A standard quadratic equation is expressed as $ax^2 + bx + c = 0$, where $a \neq 0$. Three primary algebraic methods are used to find its roots:

• Factorisation: Rewrite as $(px + q)(rx + s) = 0 \implies x = -\frac{q}{p}$ or $x = -\frac{s}{r}$.
• Completing the Square: Transform $ax^2 + bx + c = 0$ into the form $(x + p)^2 = q$. $$\text{Method: } x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0$$
• Quadratic Formula: Direct formula derived from completing the square: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

2. Forming Quadratic Equations from Roots

If $\alpha$ and $\beta$ are the roots of a quadratic equation, the equation can be constructed using:

$$\text{Sum of Roots (SOR)} = \alpha + \beta = -\frac{b}{a}$$ $$\text{Product of Roots (POR)} = \alpha\beta = \frac{c}{a}$$

The resulting quadratic equation is:

$$x^2 - (\text{SOR})x + (\text{POR}) = 0$$

3. Quadratic Inequalities

To solve inequalities such as $ax^2 + bx + c > 0$, $ax^2 + bx + c \ge 0$, $ax^2 + bx + c < 0$, or $ax^2 + bx + c \le 0$:

1. Ensure $a > 0$ by multiplying by $-1$ if necessary (reversing inequality direction).
2. Factorise to find critical values (roots $\alpha$ and $\beta$, where $\alpha < \beta$).
3. Determine solution region using Graph Method or Number Line Method:
   • For $(x - \alpha)(x - \beta) < 0 \implies \alpha < x < \beta$ (between roots).
   • For $(x - \alpha)(x - \beta) > 0 \implies x < \alpha \text{ or } x > \beta$ (outside roots).

2.2 Types of Roots of Quadratic Equations

1. The Discriminant ($b^2 - 4ac$)

The nature of the roots of $ax^2 + bx + c = 0$ is determined by its discriminant, $D = b^2 - 4ac$:

• $b^2 - 4ac > 0$: Two real and distinct roots. (Graph intersects $x$-axis at 2 distinct points).
• $b^2 - 4ac = 0$: Two real and equal roots (one repeated root). (Graph touches $x$-axis at 1 point / $x$-axis is tangent to curve).
• $b^2 - 4ac < 0$: No real roots (complex/imaginary roots). (Graph does NOT intersect or touch $x$-axis).
• $b^2 - 4ac \ge 0$: Real roots exist (covers both distinct and equal root conditions).

2.3 Quadratic Functions

1. Vertex Form and Characteristics of Graph

A quadratic function can be written in three main forms:

• General Form: $f(x) = ax^2 + bx + c$
• Factorised Form: $f(x) = a(x - p)(x - q)$, where $p$ and $q$ are $x$-intercepts.
• Vertex Form: $f(x) = a(x - h)^2 + k$, where:
  • $(h, k)$ is the vertex point (turning point).
  • $x = h$ is the axis of symmetry.
  • If $a > 0$: Curve is $U$-shaped (concave up), vertex $(h, k)$ is a minimum point, minimum value $= k$.
  • If $a < 0$: Curve is $\cap$-shaped (concave down), vertex $(h, k)$ is a maximum point, maximum value $= k$.

2. Effects of Changing Coefficients ($a, b, c$)

• Effect of $a$: Determines shape and width. $|a|$ increases $\implies$ curve becomes narrower; $|a|$ decreases $\implies$ curve becomes wider. Sign determines orientation ($a > 0 \implies \cup$, $a < 0 \implies \cap$).
• Effect of $b$: Shifts graph horizontally. Axis of symmetry is $x = -\frac{b}{2a}$.
• Effect of $c$: Determines vertical position and $y$-intercept $(0, c)$.

3. Sketching Quadratic Graphs

1. Identify shape using sign of $a$ ($a > 0 \implies \cup$, $a < 0 \implies \cap$).
2. Calculate discriminant $b^2 - 4ac$ to check $x$-intercepts.
3. Find vertex $(h, k)$ by completing the square or using $h = -\frac{b}{2a}$ and $k = f(h)$.
4. Find $y$-intercept by evaluating $f(0) = c$.
5. Find $x$-intercepts (if any) by solving $f(x) = 0$.
6. Plot critical points and draw a smooth parabola with line of symmetry $x = h$.