IGCSE Graph Families Investigation

🎯 Key idea: Do not identify a graph only by a rough picture. Use features: intercepts, symmetry, turning points, whether it crosses an axis, and whether it approaches an asymptote.

⌣ 1. Quadratic

$y=x^2$ is a quadratic. It is a U-shape, symmetric about the $y$-axis, with a minimum at $(0,0)$.

Investigation: Compare $y=x^2$, $y=x^2+3$ and $y=(x-2)^2$.

Each is quadratic. The latter two are translations of the same basic shape.

〰️ 2. Cubic and reciprocal

$y=x^3$ is cubic: it passes through the origin and has an S-shape. $y=[[frac{1}{x}]]$ is reciprocal: it has two branches and never reaches either axis.

Evidence check: $y=[[frac{1}{x}]]$ is undefined at $x=0$, so the $y$-axis is a vertical asymptote.

As $x$ becomes very large, the value approaches $0$, so the $x$-axis is a horizontal asymptote.

📈 3. Exponential

$y=2^x$ is exponential. It is always positive, crosses the $y$-axis at $(0,1)$, and grows faster as $x$ increases.

Worked example: For $y=2^x$, $x=-1,0,1,2$ give $[[frac{1}{2}]],1,2,4$.

This helps distinguish exponential growth from a straight line.

Graph investigation: Make one table for each of $x^2$, $x^3$, $[[frac{1}{x}]]$, and $2^x$. Sketch them on the same axes. Circle one feature that proves each identity.