1. Learn
  2. Quick Notes
  3. List
  4. Quadratic Functions and Equations
Back

Quadratic Functions and Equations in One Variable

1.1 Quadratic Functions and Equations
MATHEMATICS • Form 4 • Chapter 1: Quadratic Functions and Equations in One Variable

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.

\(a > 0\) — Minimum shape \(\bigcup\)
x f(x) Axis of symmetry Minimum point Root Root y-intercept
Opens upwards with a minimum point. The axis of symmetry is \(x = -\frac{b}{2a}\).
\(a < 0\) — Maximum shape \(\bigcap\)
x f(x) Axis of symmetry Maximum point Root Root y-intercept
Opens downwards with a maximum point. The axis of symmetry is \(x = -\frac{b}{2a}\).

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 \(ax^2 + bx + c = 0\) \(a \neq 0\)
1
Understand the problem
Identify known values and unknown variables such as dimensions, cost or speed.
2
Plan a strategy
Formulate an expression or function and set up a quadratic equation.
3
Implement the strategy
Expand, simplify into \(ax^2 + bx + c = 0\), and solve by factorisation. Discard non-acceptable solutions such as negative lengths.
4
Conclude and check
Answer the question in context and verify the result.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

Determine whether \(3x^2 - \frac{2}{x} + 5\) is a quadratic expression in one variable. Justify your answer.

Answer: No, because the term \(-\frac{2}{x}\) can be written as \(-2x^{-1}\), where the power of variable \(x\) is \(-1\), which is not a whole number.
2

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.

Answer: It has a maximum point because \(a = -2 < 0\). Its \(y\)-intercept is \(6\) (or coordinates \((0, 6)\)) because \(c = 6\).
3

Solve the quadratic equation \(2x^2 + 7x - 4 = 0\) by factorisation.

Answer: \((2x - 1)(x + 4) = 0 \implies 2x - 1 = 0\) or \(x + 4 = 0\). Therefore, \(x = \frac{1}{2}\) or \(x = -4\).

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

Misconception

Assuming that the axis of symmetry is always the \(y\)-axis (\(x = 0\)).

Correct Fact

The axis of symmetry is the \(y\)-axis only when \(b = 0\). When \(b \neq 0\), the axis of symmetry shifts horizontally according to \(x = -\frac{b}{2a}\).

Misconception

Thinking that a quadratic function is a one-to-one relation because it is a single function.

Correct Fact

It is a many-to-one relation because two distinct values of \(x\) (objects) can produce the exact same value of \(f(x)\) (image).

Misconception

Believing that changing \(c\) alters the width or sharpness of the parabola.

Correct Fact

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.

Feature
\(a > 0\)
Constant \(c\)
\(x = -\frac{b}{2a}\)
Roots of equation
Description
The graph has a \(\bigcup\) shape and a minimum pointCorrectTry againEmpty
Determines the \(y\)-intercept of the parabolaCorrectTry againEmpty
The equation of the axis of symmetryCorrectTry againEmpty
Values of \(x\) where the curve intersects the \(x\)-axisCorrectTry againEmpty
 

Key Terms

Quadratic expression Quadratic function Parabola Maximum point Minimum point Axis of symmetry y-intercept Roots of a quadratic equation Factorisation method
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.

Quadratic Functions and Equations in One Variable

1.1 Quadratic Functions and Equations
MATHEMATICS • Form 4 • Chapter 1: Quadratic Functions and Equations in One Variable

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.

\(a > 0\) — Minimum shape \(\bigcup\)
x f(x) Axis of symmetry Minimum point Root Root y-intercept
Opens upwards with a minimum point. The axis of symmetry is \(x = -\frac{b}{2a}\).
\(a < 0\) — Maximum shape \(\bigcap\)
x f(x) Axis of symmetry Maximum point Root Root y-intercept
Opens downwards with a maximum point. The axis of symmetry is \(x = -\frac{b}{2a}\).

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 \(ax^2 + bx + c = 0\) \(a \neq 0\)
1
Understand the problem
Identify known values and unknown variables such as dimensions, cost or speed.
2
Plan a strategy
Formulate an expression or function and set up a quadratic equation.
3
Implement the strategy
Expand, simplify into \(ax^2 + bx + c = 0\), and solve by factorisation. Discard non-acceptable solutions such as negative lengths.
4
Conclude and check
Answer the question in context and verify the result.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

Determine whether \(3x^2 - \frac{2}{x} + 5\) is a quadratic expression in one variable. Justify your answer.

Answer: No, because the term \(-\frac{2}{x}\) can be written as \(-2x^{-1}\), where the power of variable \(x\) is \(-1\), which is not a whole number.
2

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.

Answer: It has a maximum point because \(a = -2 < 0\). Its \(y\)-intercept is \(6\) (or coordinates \((0, 6)\)) because \(c = 6\).
3

Solve the quadratic equation \(2x^2 + 7x - 4 = 0\) by factorisation.

Answer: \((2x - 1)(x + 4) = 0 \implies 2x - 1 = 0\) or \(x + 4 = 0\). Therefore, \(x = \frac{1}{2}\) or \(x = -4\).

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

Misconception

Assuming that the axis of symmetry is always the \(y\)-axis (\(x = 0\)).

Correct Fact

The axis of symmetry is the \(y\)-axis only when \(b = 0\). When \(b \neq 0\), the axis of symmetry shifts horizontally according to \(x = -\frac{b}{2a}\).

Misconception

Thinking that a quadratic function is a one-to-one relation because it is a single function.

Correct Fact

It is a many-to-one relation because two distinct values of \(x\) (objects) can produce the exact same value of \(f(x)\) (image).

Misconception

Believing that changing \(c\) alters the width or sharpness of the parabola.

Correct Fact

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.

Feature
\(a > 0\)
Constant \(c\)
\(x = -\frac{b}{2a}\)
Roots of equation
Description
The graph has a \(\bigcup\) shape and a minimum pointCorrectTry againEmpty
Determines the \(y\)-intercept of the parabolaCorrectTry againEmpty
The equation of the axis of symmetryCorrectTry againEmpty
Values of \(x\) where the curve intersects the \(x\)-axisCorrectTry againEmpty
 

Key Terms

Quadratic expression Quadratic function Parabola Maximum point Minimum point Axis of symmetry y-intercept Roots of a quadratic equation Factorisation method
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Function and Quadratic Equation in One Variable
Topic : Quadratic Functions and Equations
Form 4 Mathematics
Related notes

Report this note
© 2026 Pandai.org All Rights Reserved

Quiz
View video m
Notes
Account