Chapter 1: Quadratic Functions and Equations in One Variable

Overview

Welcome to Chapter 1 of Form 4 Mathematics! This chapter introduces you to the fascinating world of quadratic functions and equations. You'll learn how to identify quadratic expressions, graph parabolas, find roots of quadratic equations, and solve real-world problems involving quadratic relationships.

What You'll Learn:

Identify and understand quadratic expressions and functions
Graph quadratic functions and analyze their properties
Find roots using various methods including factorization
Apply quadratic concepts to solve practical problems

Learning Objectives

After completing this chapter, you will be able to:

Identify the characteristics of quadratic expressions in one variable
Recognize quadratic functions as many-to-one relationships
Describe the characteristics of quadratic functions
Investigate and generalize the effects of changing values a, b, and c in f(x) = a $x^2$ + bx + c
Explain the meaning of roots and determine them using factorization
Sketch graphs of quadratic functions
Solve problems involving quadratic equations

Key Concepts

General Form of Quadratic Expressions

A quadratic expression is an expression in the form $ax^2 + bx + c$ , where:

a, b, c are constants
a ≠ 0
x is the variable
The highest power of the variable is 2

Example: $3x^2 + 5x - 2$ is a quadratic expression where a = 3, b = 5, c = -2

Quadratic Functions Visualized

Properties of Quadratic Functions

The following diagram shows the key characteristics of quadratic functions:

Graph Shape and Properties

The graph of a quadratic function is a smooth, symmetrical curve known as a parabola.

Effects of the Constant 'a':

When a > 0:

Graph opens upwards like a "smile" (∪)
Has one minimum point (vertex)
Example: f(x) = $x^2$

When a < 0:

Graph opens downwards like a "frown" (∩)
Has one maximum point (vertex)
Example: f(x) = - $x^2$

Effects of the Constants:

Constant	Effect on Graph
a	Controls width and direction of opening
b	Determines the axis of symmetry position
c	Determines the y-intercept position

Roots of Equations

The roots of a quadratic equation $ax^2 + bx + c = 0$ are the values of x that satisfy the equation. These roots represent the x-intercepts where the graph crosses the x-axis.

Discriminant Analysis

The discriminant determines the nature of the roots and the graph behavior:

Vertex Form and Transformations

Quadratic functions can be written in vertex form, which makes transformations easier to understand:

f(x) = a(x - h)^2 + k

where (h, k) is the vertex.

Important Formulas and Methods

1. Factorization Method

The main method for finding roots of quadratic equations. The equation a $x^2$ + bx + c = 0 is expressed in the form (px + r)(qx + s) = 0, giving roots x = -r/p and x = -s/q.

Example: Solve $x^2$ - 5x + 6 = 0

Solution: $x^2$ - 5x + 6 = 0 (x - 2)(x - 3) = 0 Therefore, x = 2 or x = 3

2. Axis of Symmetry Formula

The axis of symmetry for the quadratic function graph is given by:

x = -\frac{b}{2a}

This formula is used to find the x-coordinate of the minimum or maximum point.

Example: For f(x) = 2 $x^2$ - 4x + 1 x = -(-4)/(2 × 2) = 4/4 = 1

3. Vertex Formula

The vertex (turning point) coordinates are:

\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)

4. Discriminant

The discriminant determines the nature of the roots:

\Delta = b^2 - 4ac

Δ > 0: Two distinct real roots
Δ = 0: One real root (repeated)
Δ < 0: No real roots (complex roots)

Graph Sketching Process

Follow these steps to sketch quadratic graphs:

Graph Sketching Process

Follow these steps to sketch quadratic graphs:

Determine the graph shape (based on the sign of a)
Find the y-intercept (value of c)
Find x-intercepts/roots (solve f(x) = 0)
Find the vertex/minimum/maximum point (using axis of symmetry)
Plot additional points if needed for accuracy

Example: Graph Sketching with Steps

Let's visualize the process for sketching $f(x) = x^2 - 4x + 3$ :

Example: Sketch the graph of f(x) = $x^2$ - 4x + 3

Shape: a = 1 > 0, opens upwards (∪)
Y-intercept: When x = 0, f(0) = 3
X-intercepts: $x^2$ - 4x + 3 = 0 → (x - 1)(x - 3) = 0 → x = 1, x = 3
Vertex: x = -(-4)/(2 × 1) = 2, f(2) = 4 - 8 + 3 = -1 Vertex at (2, -1)

Step-by-Step Solved Examples

Example 1: Finding Roots

Problem: Find the roots of the equation 2 $x^2$ + 5x - 3 = 0

Solution: Using factorization: 2 $x^2$ + 5x - 3 = 0 (2x - 1)(x + 3) = 0

Therefore: 2x - 1 = 0 → x = 1/2 x + 3 = 0 → x = -3

Answer: The roots are x = 1/2 and x = -3

Example 2: Graph Sketching

Problem: Sketch the graph of f(x) = $x^2$ - 6x + 8

Solution:

Shape: a = 1 > 0, opens upwards
Y-intercept: f(0) = 8
X-intercepts: $x^2$ - 6x + 8 = 0 (x - 2)(x - 4) = 0 → x = 2, x = 4
Vertex: x = -(-6)/(2 × 1) = 3 f(3) = 9 - 18 + 8 = -1 Vertex at (3, -1)

Plot the key points:

Y-intercept: (0, 8)
X-intercepts: (2, 0) and (4, 0)
Vertex: (3, -1)

Example 3: Real-world Application

Problem: A ball is thrown upwards from a height of 2 meters with an initial velocity of 14 m/s. The height h (in meters) after t seconds is given by h(t) = -5 $t^2$ + 14t + 2. Find when the ball hits the ground.

Solution: The ball hits the ground when h(t) = 0: -5 $t^2$ + 14t + 2 = 0 Multiply by -1: 5 $t^2$ - 14t - 2 = 0

Using quadratic formula: t = [14 ± √(196 + 40)] / 10 = [14 ± √236] / 10 t = [14 ± 15.36] / 10

t = 2.936 or t = -0.136

Since time cannot be negative, t = 2.94 seconds

Answer: The ball hits the ground after approximately 2.94 seconds.

Real-world Applications

Quadratic functions appear in many real-world situations:

Projectile Motion Analysis

Business and Economics Applications

Engineering Applications

Real-world Applications

Quadratic functions appear in many real-world situations:

1. Projectile Motion

Path of thrown objects
Basketball trajectories
Firework explosions

2. Business and Economics

Profit maximization
Cost analysis
Revenue modeling

3. Engineering

Bridge design
Parabolic antennas
Suspension cables

4. Physics

Optics (lens shapes)
Orbital mechanics
Wave patterns

Mathematical Proofs

Proof of the Quadratic Formula

Starting with a $x^2$ + bx + c = 0:

Divide by a: $x^2$ + (b/a)x + c/a = 0
Complete the square: $x^2$ + (b/a)x + (b/2a)² = (b/2a)² - c/a
(x + b/2a)² = $b^2$ /4 $a^2$ - c/a
(x + b/2a)² = ( $b^2$ - 4ac)/4 $a^2$
x + b/2a = ±√( $b^2$ - 4ac)/2a
x = -b/2a ± √( $b^2$ - 4ac)/2a

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Summary Points

Quadratic Expression: $ax^2 + bx + c$ (a ≠ 0)
Quadratic Function: $f(x) = ax^2 + bx + c$
Quadratic Equation: $ax^2 + bx + c = 0$
Root: Value of variable that satisfies the equation
Axis of Symmetry: Line dividing parabola into two equal parts
Vertex: Turning point of the parabola
Y-intercept: Point where graph crosses y-axis

Key Formula Summary

Summary Points

Quadratic Expression: $ax^2 + bx + c$ (a ≠ 0)
Quadratic Function: $f(x) = ax^2 + bx + c$
Quadratic Equation: $ax^2 + bx + c = 0$
Root: Value of variable that satisfies the equation
Axis of Symmetry: Line dividing parabola into two equal parts
Vertex: Turning point of the parabola
Y-intercept: Point where graph crosses y-axis

Practice Tips for SPM Students

1. Master Factorization

Practice different factoring techniques
Learn to spot perfect squares
Practice with various coefficient values

2. Understand Graph Properties

Memorize the effects of a, b, and c
Practice sketching graphs without a calculator
Learn to identify key features quickly

3. Problem-solving Strategies

Read the problem carefully
Identify what is being asked
Choose the appropriate method
Show all working steps

4. Exam Preparation

Practice past year questions
Time yourself for Paper 1
Understand the marking scheme
Show all working for Paper 2

5. Common Mistakes to Avoid

Forgetting that a ≠ 0 for quadratic expressions
Confusing the axis of symmetry formula
Making calculation errors in factorization
Not checking the reasonableness of answers

SPM Exam Tips

Paper 1 (Multiple Choice)

Look for clues in the question wording
Use elimination method for difficult questions
Practice mental calculation for speed
Double-check your answers

Paper 2 (Structured)

Show all working steps clearly
Label your graphs properly
Use graph paper for sketching
Write units for real-world problems

Did You Know? The study of quadratic equations dates back to ancient Babylon around 2000 BC! The Babylonians had methods for solving quadratic problems that were remarkably similar to our modern approaches.

Next Chapter: As you proceed to Chapter 2, you'll discover the fascinating world of number bases and how different number systems work together in computer science and mathematics.

Chapter 2: Number Bases