Reversing transformations, sketching transformed graphs and inverses

Last time

Before we start: last time, in Describing combined transformations, equations, ranges and inverses, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Read the graph backwards

A picture of a transformed graph carries a code. The period, the height, the starting point: each tells you one transformation. Read it off, and write the equation. Or go the other way: sketch the new graph, or its mirror image the inverse, by moving a handful of key points.

Sketching with key points

To sketch a transformed graph, track the key points: ends, turning points, and intercepts. Move each one, then join them with the same kind of curve. In y equals 2 f of x, the heights double. In y equals f of x over 2, the widths double. Apply the rule to each coordinate. Label the coordinates of the key points on your sketch.

A zigzag redrawn

A graph f goes from 0 to 2 as a zigzag. The transformed graph g has bottom points at height 2, and top points at height 4. Plot the five points: 0, 2; 0.5, 4; 1, 2; 1.5, 4; and 2, 2. Join them with straight lines, because f had straight lines. Then write g in terms of f. Compare heights and widths, and read off the stretch and translation.

A past-paper example

October November 2022, Paper 12, question 5. The graph of y equals f of x is given. Sketch y equals g of x and find g in terms of f. The sketch is five points joined by straight lines: the bottom points at y equals 2 and the top points at 4. Then express g using a stretch and a translation of f.