Graph transformations
Pick a graph, then stretch, reflect and slide it with y = a f(b(x − h)) + k. The faint curve is the original, the bold one is the image, and the gold point shows exactly where one point goes.
Readouts
What's happening
Changes outside the bracket act on y and do what they look like: y = a f(x) stretches the graph parallel to the y-axis by scale factor a (a negative a also reflects it in the x-axis), and y = f(x) + k moves it up by k. Changes inside the bracket act on x and do the opposite of what they look like: y = f(bx) stretches parallel to the x-axis by scale factor 1/b, so f(2x) squashes the graph to half its width, and f(x − h) moves it right by h, so f(x + 2) moves it left by 2. Put together, every point (x, y) on y = f(x) goes to (x/b + h, ay + k). Order matters inside the bracket: f(2x − 6) is f(2(x − 3)), a stretch by 1/2 and then a move of 3 to the right, not 6. Asymptotes move with the graph, so 1/(x − 3) + 1 has asymptotes x = 3 and y = 1.
GCSE Maths (AQA, Edexcel, OCR) Higher: translations and reflections of graphs, y = f(x) + a, f(x + a), −f(x) and f(−x). A-Level Maths: stretches y = af(x) and y = f(ax), combined transformations and asymptotes.
Work through the numbers with Graphing Calculator.
Challenge
Predict first: the point (2, 4) lies on y = x². Where does it go on y = 2(x + 1)² − 3? Work out the new x-coordinate, type it in, press Check, then set a = 2, h = −1 and k = −3 with the point at x = 2.
Here a = 2, b = 1, h = −1 and k = −3, so (2, 4) goes to (2 − 1, 2 × 4 − 3) = (1, 5). The graph is stretched parallel to the y-axis by scale factor 2 and then translated by the vector (−1, −3).
FAQ
- Why does f(x + 2) move the graph left?
- Because the new graph reaches each value 2 earlier: whatever f did at x = 0 now happens at x = −2. So y = f(x + a) is a translation by the vector (−a, 0).
- What is the difference between 2f(x) and f(2x)?
- 2f(x) doubles every y-value: a stretch parallel to the y-axis, scale factor 2. f(2x) reaches each value at half the x: a stretch parallel to the x-axis, scale factor 1/2.
- What do −f(x) and f(−x) do?
- −f(x) is a reflection in the x-axis (every y changes sign). f(−x) is a reflection in the y-axis (every x changes sign).
- How do I describe y = f(2x − 6)?
- Factor the bracket first: f(2x − 6) = f(2(x − 3)). That is a stretch parallel to the x-axis with scale factor 1/2, then a translation by (3, 0). Translating by 6 first would give the wrong graph.