1.1 Quadratic Expressions and Functions
General Form of a Quadratic Expression
A quadratic expression in one variable is an algebraic expression with a single variable where the highest power of the variable is 2. The general form is:
$$ax^2 + bx + c$$
where $a$, $b$, and $c$ are constants and $a \neq 0$. The variable is $x$.
Characteristics of Quadratic Functions
A quadratic function is written as $f(x) = ax^2 + bx + c$. Its graphical representation is a smooth curve called a parabola.
- Shape of the Graph:
- If $a > 0$, the graph opens upwards $\cup$ (minimum point).
- If $a < 0$, the graph opens downwards $\cap$ (maximum point).
- Effect of Coefficient $a$: Controls the width of the opening. Larger $|a|$ produces a narrower curve; smaller $|a|$ produces a wider curve.
- Effect of Coefficient $b$: Controls the position of the axis of symmetry relative to the y-axis.
- When $a$ and $b$ have the same sign ($a \cdot b > 0$), the axis of symmetry lies to the left of the y-axis.
- When $a$ and $b$ have opposite signs ($a \cdot b < 0$), the axis of symmetry lies to the right of the y-axis.
- When $b = 0$, the axis of symmetry is the y-axis itself ($x = 0$).
- Effect of Constant $c$: Determines the y-intercept of the graph, which occurs at $(0, c)$.
Axis of Symmetry and Extremum Points
The line of symmetry divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by:
$$x = -\frac{b}{2a}$$
The maximum or minimum turning point lies directly on this axis of symmetry at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.
1.2 Roots of a Quadratic Equation
Quadratic Equations
A quadratic equation in one variable is formed when a quadratic expression is equated to zero:
$$ax^2 + bx + c = 0$$
Roots of Quadratic Equations
The roots of a quadratic equation are the values of $x$ that satisfy the equation $ax^2 + bx + c = 0$. Graphically, the roots represent the x-intercepts of the graph $f(x) = ax^2 + bx + c$.
Methods to Determine Roots
- Factorisation Method: Express $ax^2 + bx + c = 0$ in the factored form $(px + q)(rx + s) = 0$, yielding roots $x = -\frac{q}{p}$ and $x = -\frac{s}{r}$.
- Graphical Method: Identify the points where $y = f(x)$ crosses the x-axis ($y = 0$).
- Quadratic Formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
1.3 Sketching Graphs of Quadratic Functions
To sketch the graph of $f(x) = ax^2 + bx + c$, follow these essential steps:
- Determine the shape ($\cup$ if $a > 0$, $\cap$ if $a < 0$).
- Calculate the y-intercept at $(0, c)$.
- Find the x-intercepts (roots) by solving $f(x) = 0$. If $b^2 - 4ac < 0$, there are no real x-intercepts.
- Calculate the coordinates of the turning point (maximum or minimum point) using $x = -\frac{b}{2a}$.
- Plot the key points and join them with a smooth parabolic curve.