1.1 Quadratic Functions and Equations
Quadratic expressions, functions, and equations in one variable form the foundation of non-linear algebra. In daily life, curved paths such as the trajectory of a kicked football form a parabola, which can be modelled using quadratic expressions.
Learning Objectives
- Identify and describe the characteristics of quadratic expressions in one variable.
- Recognise quadratic functions as many-to-one relations and describe their characteristics.
- Investigate and make generalisations about the effects of changing the values of \(a\), \(b\), and \(c\) on graphs of quadratic functions \(f(x) = ax^2 + bx + c\).
- Form quadratic functions based on situations and relate them to quadratic equations.
- Explain the meaning of the roots of a quadratic equation and determine them by graphical and factorisation methods.
- Sketch graphs of quadratic functions under different conditions.
- Solve real-world problems involving quadratic equations.
Characteristics and Key Anatomy of a Quadratic Graph
See how the constants \(a\), \(b\), and \(c\) control the shape, turning point, axis of symmetry, roots and \(y\)-intercept of a parabola.
Brief Explanation
What is a quadratic expression?
A quadratic expression in one variable is an algebraic expression where the highest power of the variable is 2. Its general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, \(a \neq 0\), and \(x\) is the variable.
Why must \(a \neq 0\)?
If \(a = 0\), the \(x^2\) term disappears and the expression becomes linear. The powers of the variable must also be whole numbers; terms such as \(x^{-2}\) or \(x^{\frac{1}{2}}\) are not quadratic.
| Form | General notation | Key condition |
|---|---|---|
| Quadratic expression | \(ax^2 + bx + c\) | \(a \neq 0\) |
| Quadratic function | \(f(x) = ax^2 + bx + c\) | \(a \neq 0\) |
| Quadratic equation | \(ax^2 + bx + c = 0\) | \(a \neq 0\) |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Determine whether \(3x^2 - \frac{2}{x} + 5\) is a quadratic expression in one variable. Justify your answer.
Given the quadratic function \(f(x) = -2x^2 + 4x + 6\), state whether the stationary point is a maximum or minimum point, and determine its \(y\)-intercept.
Solve the quadratic equation \(2x^2 + 7x - 4 = 0\) by factorisation.
Common Mistakes
- Treating expressions with negative exponents such as \(x^{-2}\) or square roots such as \(x^{\frac{1}{2}}\) as quadratic expressions simply because an \(x^2\) term is also present.
- Confusing a quadratic expression \(ax^2 + bx + c\), a quadratic function \(f(x) = ax^2 + bx + c\), and a quadratic equation \(ax^2 + bx + c = 0\).
- Forgetting to rearrange equations into standard form \(ax^2 + bx + c = 0\) before factorising or identifying the values of \(a\), \(b\), and \(c\). For example, in \(1 - x - 2x^2\), \(a = -2\), \(b = -1\), and \(c = 1\).
- Accepting negative solutions in real-life geometric problems where lengths, widths, or ages cannot be negative.
Misconceptions
Assuming that the axis of symmetry is always the \(y\)-axis (\(x = 0\)).
The axis of symmetry is the \(y\)-axis only when \(b = 0\). When \(b \neq 0\), the axis of symmetry shifts horizontally according to \(x = -\frac{b}{2a}\).
Thinking that a quadratic function is a one-to-one relation because it is a single function.
It is a many-to-one relation because two distinct values of \(x\) (objects) can produce the exact same value of \(f(x)\) (image).
Believing that changing \(c\) alters the width or sharpness of the parabola.
Changing \(c\) only shifts the entire graph vertically up or down, directly changing the \(y\)-intercept, without changing the shape or width.
Summary
- Quadratic expression: \(ax^2 + bx + c\) with highest power 2 and \(a \neq 0\).
- Quadratic function: \(f(x) = ax^2 + bx + c\); forms a parabola and represents a many-to-one relation.
- Role of \(a\): Determines graph orientation (\(a > 0 \rightarrow \bigcup\) minimum point; \(a < 0 \rightarrow \bigcap\) maximum point) and width (smaller \(|a| \rightarrow\) wider curve).
- Role of \(b\): Determines the position of the axis of symmetry \(x = -\frac{b}{2a}\).
- Role of \(c\): Determines the \(y\)-intercept at \((0, c)\).
- Roots: The values of \(x\) satisfying \(ax^2 + bx + c = 0\), identical to the \(x\)-intercepts of \(f(x) = ax^2 + bx + c\).
- Factorisation: Rearrange to \(ax^2 + bx + c = 0\), factorise into \((px + q)(rx + s) = 0\), and solve for \(x\).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following algebraic expressions is a quadratic expression in one variable?
2 What is the equation of the axis of symmetry for the quadratic function \(f(x) = 2x^2 + 4x - 3\)?
B. Fill in the Blanks
3 For a quadratic function \(f(x) = ax^2 + bx + c\), the turning point is a minimum point when \(a\) is than zero.
4 The roots of a quadratic equation \(ax^2 + bx + c = 0\) correspond to the -intercepts on the graph of the function \(f(x) = ax^2 + bx + c\).
C. Drag and Drop: Match Each Feature to Its Description
Drag each feature to the correct description. On a phone, tap the answer choice first, then tap the description box.
