A-level · Beta

Functions and graph transformations

Move and stretch graphs; compare linear inverse and composite functions.

Drag T to translate the graph. Drag Q to trace an input and its output.

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

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

The grey dashed graph is the original; the blue graph is its transformation.

For transformations, multiplying inside f changes the horizontal scale. Linear inverse and composite views use f(x) = ax + h and g(x) = bx + k.

Predict, test and explain

Step 1 of 3

Predict

Compared with y = f(x), which way does y = f(x − 2) move?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Compared with y = f(x), which way does y = f(x − 2) move?

$y=f(x-2)$

Hint

An inside shift is found by solving the bracket equal to the original input.

Reveal answer

An original input of zero now occurs at x = 2.

$\text{right }2$

Hint 1

An inside shift is found by solving the bracket equal to the original input.

Hint 2

A horizontal multiplier b changes widths by the reciprocal factor.

Hint 3

For linear inverse and composite views, f(x) = ax + h and g(x) = bx + k.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the turning point of y = 2(x − 3)² − 1.

Answer: Turning point (3, −1).

Connect this idea

Watch out: An inside minus sign does not mean a shift left: x − 2 = 0 occurs at x = 2.

Specification and learning route

AQA B8, B9 · Edexcel Pure 2.8, Pure 2.9

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Function notation; Common graphs; Coordinates.