Functions and graphs

Graph transformations become clearer when you can move them yourself.

This prototype lets learners compare a base function with a transformed version, adjusting vertical scale, horizontal scale and translations in real time.

The aim is to turn graph transformations from rules to memorise into patterns students can predict, test and explain.

Graph transformations

Compare f(x) with a transformed graph

Start with a familiar function, then adjust vertical scale, horizontal scale and translations. The grey graph is f(x); the blue graph is the transformed function.

Base function

f(x) = x^2

g(0)

g(1)

Controls

Base function

a: 1.00

b: 1.00

c: 0.00

d: 0.00

Presets

g(x) = 1 f(1(x - 0)) + 0

No transformation from f(x).

Domain note: All real values of x.

What this shows

Changes outside f affect the output, so they act vertically. Changes inside f affect the input, so they act horizontally.

Horizontal transformations often feel backwards at first: x - 2 shifts the graph right, not left.

The same transformation language works across many functions, not just quadratics.

Guided tasks

Build a graph that is shifted right 2 and up 3. Which parameters changed?

Reflect the graph in the x-axis, then in the y-axis. How are the controls different?

Compare b = 2 and b = 0.5. Why is this called a horizontal scale factor of 1 divided by b?

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can become a core graphing and functions tool.

Add point mapping so learners can see where key points move.

Add challenge mode: match the target transformed graph.

Add asymptote-aware examples such as reciprocal and exponential functions.

Add teacher prompts for GCSE and A Level graph transformation lessons.