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O Level · Mathematics 0580

Quadratic Graphs

Revise Quadratic Graphs for Mathematics 0580 (O Level) — revision notes, 109 practice questions and instant AI marking.

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What is a Quadratic Graph?

Definition

y = ax² + bx + c

where a ≠ 0

A quadratic graph is a smooth curve that displays a special relationship between variables. The equation contains a squared term (x²) as its highest power.

Key Characteristics

Shape: The curve is called a parabola
Symmetry: Has a vertical line of symmetry
Y-intercept: Always crosses the y-axis at (0, c)
X-intercepts: Can cross the x-axis 0, 1, or 2 times (roots)
Turning point: Has either a minimum or maximum point (vertex)

U-Shape vs N-Shape

Positive Quadratic (a > 0)

When the coefficient of x² is positive:

Creates a U-shaped curve
Has a minimum point
Opens upward

Negative Quadratic (a < 0)

When the coefficient of x² is negative:

Creates an N-shaped curve
Has a maximum point
Opens downward

How to Sketch a Quadratic Graph

Step-by-Step Method

Sketch the axes - Draw your x and y axes
Find the y-intercept - It's the constant term (c) in y = ax² + bx + c
Find the roots - Solve ax² + bx + c = 0
Determine the shape - Check if a is positive (∪) or negative (∩)
Sketch the curve - Draw a smooth curve through your points
Mark turning point - Show the vertex coordinates if known

Finding the Y-Intercept

Substitute x = 0 into y = ax² + bx + c

This gives y = c, so the y-intercept is at (0, c)

Finding the Roots (X-Intercepts)

Roots are found by solving y = 0, or ax² + bx + c = 0

Methods to find roots:

Factorising the quadratic
Completing the square
Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Interactive Tool: Sketch Generator

Enter a quadratic equation:

a (coefficient of x²):

b (coefficient of x):

c (constant):

Finding the Turning Point

Method 1: Completing the Square

Standard form: y = a(x - p)² + q

Turning point coordinates: (p, q)

⚠️ Important Sign Rule:

y = (x - 3)² + 2 → vertex at (3, 2)
y = (x + 3)² + 2 → vertex at (-3, 2)

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Also in the full note

Finding the Equation from a Graph
Worked Examples
Method 2: Using Differentiation
Interactive Tool: Find Turning Point
When You Know the Vertex and One Point
When You Know the Roots and One Point
Special Case: If a = 1
Interactive Tool: Find Equation

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Quadratic Graphs

What is a Quadratic Graph?

Definition

Key Characteristics

U-Shape vs N-Shape

Positive Quadratic (a > 0)

Negative Quadratic (a < 0)

How to Sketch a Quadratic Graph

Step-by-Step Method

Finding the Y-Intercept

Finding the Roots (X-Intercepts)

Interactive Tool: Sketch Generator

Enter a quadratic equation:

Finding the Turning Point

Method 1: Completing the Square

Past papers that test Quadratic Graphs

More Mathematics topics