Higher Mathematics

Domain, range and transformations

State sensible domains and ranges, then sketch translations, reflections and stretches.

Topic practice Formula support Mixed revision

Before you start

Read f(x) notation as an input-output rule.
Know the parent graph before applying a transformation.
Use brackets carefully for changes inside the function.

Higher Mathematics lesson

Explanation

Transformations describe how the graph of y = f(x) moves or changes shape. Output changes such as f(x) + a move the graph vertically; input changes such as f(x − a) move it horizontally.

Domain tells you the allowed inputs. Range tells you the possible outputs. At Higher, graph questions often ask you to connect these restrictions to transformed graphs.

Visual support

Method and rules

f(x) + a shifts the graph up a units.
f(x − a) shifts the graph right a units.
af(x) stretches outputs by scale factor a.
f(ax) changes inputs; horizontal scale factors work in the opposite direction.

Worked examples

Worked example 1

Describe a transformation

Describe y = f(x − 3) + 2 from y = f(x)

The x − 3 is inside the function, so move the graph right 3.
The + 2 is outside the function, so move the graph up 2.

Answer: The graph is translated 3 right and 2 up.

Worked example 2

Evaluate after a transformation

If f(4) = 7, find the matching point on y = f(x − 2).

For f(x − 2), the same output occurs when x − 2 = 4.
Solve x = 6.
Keep the output 7.

So: The point becomes (6, 7).

Watch out

Moving f(x − a) left instead of right.
Confusing input changes with output changes.
Writing f(x) + 3 as f(x + 3).
Forgetting that domain and range may change after a transformation.

Exam reminder

In graph transformation questions, name the transformation and give exact coordinates for key points when asked. Horizontal changes are the common trap.