Transformations of graphs

Transformations of graphs move, stretch or flip the graph of y = f(x) by changing its equation. Adding outside the function, f(x) + a, moves the graph up; adding inside, f(x + a), moves it left. Multiplying outside, af(x), stretches it vertically; multiplying inside, f(ax), squashes it horizontally. A minus sign outside reflects it in the x-axis, and inside reflects it in the y-axis.

Transformations of graphs summary

At GCSE, AQA's specification (A13) asks you to sketch translations and reflections of a given function. At A-level, AQA adds stretches and combinations of transformations. The rule to remember: changes outside the bracket affect y and do what you expect; changes inside the bracket affect x and do the opposite.

Translations

y = f(x) + a: up or down

Adding a to every output moves every point up by a. For f(x) = x², the graph of y = x² + 3 is the same parabola with its vertex at (0, 3). As a vector, the translation is (0, a).

y = f(x + a): left or right

This is the one that catches people out. In y = f(x + 2), the graph reaches each y-value when x is 2 smaller than before, so it moves 2 to the left. y = f(x − 2) moves it 2 to the right. As a vector, f(x + a) is a translation by (−a, 0).

Both together give a translation by a vector: y = (x − 3)² + 1 is y = x² moved 3 right and 1 up, so its vertex moves from (0, 0) to (3, 1).

Reflections

• y = −f(x) reflects the graph in the x-axis. Every y-value changes sign, so the parts above the axis go below it.
• y = f(−x) reflects the graph in the y-axis. Every point swaps to the other side, (x, y) to (−x, y).

Some graphs don't change. y = x² is symmetrical about the y-axis, so f(−x) gives the same graph. That is a quick check on an answer.

Stretches (A-level)

• y = af(x) is a vertical stretch with scale factor a, parallel to the y-axis. Every y-value is multiplied by a; points on the x-axis stay put.
• y = f(ax) is a horizontal stretch with scale factor 1/a, parallel to the x-axis. Every x-value is divided by a, so y = f(2x) squashes the graph to half its width. Points on the y-axis stay put.

Now try one: identify a transformation from its equation, then press Mark to check it in the Graphing Functions course.

Worked examples

Example 1: where does a point go?

The point (4, −2) is on the graph of y = f(x). Find where it moves to on each of these graphs.

Example 2: a turning point

The lowest point of the curve y = f(x) is (1, −4). Write down the minimum point of y = f(x − 3) + 2. The graph moves 3 right and 2 up, so the minimum moves to (4, −2).

Example 3: an asymptote

The graph of y = 1/x has asymptotes x = 0 and y = 0. For y = 1/(x + 2) − 1, the graph moves 2 left and 1 down, so the asymptotes move to x = −2 and y = −1. Asymptotes move with the graph, so always transform them too.

Example 4: combining transformations (A-level)

Describe how y = 2f(x) + 1 is obtained from y = f(x). Work from the inside out: first a vertical stretch with scale factor 2, then a translation up 1. Doing them the other way round would give 2(f(x) + 1) = 2f(x) + 2, a different graph, so the order of vertical changes matters.

Check any of these by plotting both equations in the graphing calculator. The same rules apply to trig graphs, so y = sin(x + 90°) is y = sin x moved 90° left, as the trig graphs guide shows.

Common mistakes

• Moving the wrong way for f(x + a). f(x + 2) moves the graph left, not right.
• Using a instead of 1/a for f(ax). y = f(3x) has scale factor ⅓, squashing the graph towards the y-axis.
• Mixing up the reflections. −f(x) flips in the x-axis; f(−x) flips in the y-axis.
• Forgetting asymptotes and intercepts. Transform these key features as well as the curve, and label them.
• Describing a translation without a vector. At A-level, say "translation by the vector (0, 3)", not just "moves up".

To practise graphs in a short timed paper, try today's GCSE Maths Higher or A-level Maths: Algebra & Functions paper.

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