Graphs of Equations and Functions — OCR GCSE Study Guide

## 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.