Graphs of Equations and Functions

Overview

Welcome to the visual side of Mathematics! Graphs of Equations and Functions form a cornerstone of the GCSE Mathematics specification, translating abstract algebra into visual shapes. This topic is crucial because it connects multiple areas of mathematics—from solving simultaneous equations to understanding real-world rates of change. Examiners love testing this topic as it assesses your ability to process information algebraically and visually simultaneously.

Whether you are sketching a simple linear graph or finding the roots of a complex quadratic, the skills you develop here will earn you significant marks. Questions range from straightforward 'complete the table and plot' tasks to complex multi-step problems involving tangents and intersections.

Listen to the companion podcast below to reinforce your learning on the go:

Key Concepts

Concept 1: Linear Graphs (Straight Lines)

Linear graphs represent equations with no powers higher than 1 (e.g., y = 2x + 1). They always form straight lines. The general equation is y = mx + c.

• Gradient (m): This is the steepness of the line. It tells you how much y increases for every 1 unit increase in x. A positive gradient goes uphill (left to right), while a negative gradient goes downhill.
• y-intercept (c): This is the exact point where the line crosses the y-axis (where x = 0).

Why it works: The gradient is a constant rate of change. Because it never changes, the line never curves.

Example: For y = -3x + 4, the gradient is -3 (for every 1 step right, go 3 steps down), and it crosses the y-axis at (0, 4).

Concept 2: Quadratic Graphs (Parabolas)

Quadratic graphs represent equations where the highest power of x is 2 (e.g., y = x^2 - 4x + 3). They form symmetrical 'U' or 'n' shaped curves called parabolas.

• Roots (x-intercepts): Where the curve crosses the x-axis. These are the solutions to the equation when y = 0.
• y-intercept: Where the curve crosses the y-axis (when x = 0).
• Turning Point: The lowest point (minimum) for a U-shape, or highest point (maximum) for an n-shape.
• Axis of Symmetry: The vertical line passing exactly through the turning point.

Example: For y = x^2 - 9, the roots are at x = 3 and x = -3, the y-intercept is at (0, -9), and the turning point is also at (0, -9).

Concept 3: Higher Tier Graphs

Higher tier candidates must master three additional graph types:

**Cubic Graphs (y = ax^3 + bx^2 + cx + d)**These form an 'S' shape. They can cross the x-axis up to three times and may have two turning points or a single point of inflection.

**Reciprocal Graphs (y = \frac{k}{x})**These graphs have two separate branches. They feature asymptotes—lines the curve approaches but never touches. For y = \frac{1}{x}, the asymptotes are the x-axis (y=0) and y-axis (x=0).

**Exponential Graphs (y = a^x)**These model rapid growth or decay. They curve upwards steeply and have an asymptote along the x-axis.

**Circle Graphs (x^2 + y^2 = r^2)**This equation produces a perfect circle centred at the origin (0,0) with a radius of r.

Mathematical Relationships

Gradient Formula: m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
Use this to find the gradient between any two points (x_1, y_1) and (x_2, y_2).

Circle Equation: x^2 + y^2 = r^2
If x^2 + y^2 = 36, the radius is \sqrt{36} = 6.

Practical Applications

Graphs aren't just abstract curves. Linear graphs model fixed costs (like a taxi fare with a base charge plus a rate per mile). Exponential graphs model compound interest, population growth, and radioactive decay. Quadratics model the trajectory of thrown objects (projectiles) under gravity.