The point (1, −2) lies on y = f(x). Find its image on y = 3f(x − 2) + 1.
- Inside the bracket, x − 2 means a translation of 2 to the right: x = 1 + 2 = 3.
- Outside, multiply y by 3: 3 × (−2) = −6.
- Then add 1: −6 + 1 = −5.
Answer: The image is (3, −5).
Transformations of graphs
Maths: Analysis and Approaches · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
A transformation moves or reshapes the graph of y = f(x) without changing its basic form. There are three kinds: a translation (a slide), a reflection (a flip) and a stretch.
A change made outside the function, such as f(x) + 4 or 3f(x), acts on the y-values and does exactly what it looks like. A change made inside the bracket, such as f(x − 3) or f(2x), acts on the x-values and does the opposite of what it looks like: f(x − 3) moves the graph 3 to the right, and f(2x) halves every x-coordinate.
To find the image of a point, change its x-coordinate using the inside changes and its y-coordinate using the outside changes.
Rules to remember
- y = f(x) + b: translation b units up (down if b is negative); (x, y) → (x, y + b)
- y = f(x − a): translation a units to the right; y = f(x + a): a units to the left
- y = −f(x): reflection in the x-axis; (x, y) → (x, −y)
- y = f(−x): reflection in the y-axis; (x, y) → (−x, y)
- y = pf(x): vertical stretch with scale factor p; (x, y) → (x, py)
- y = f(qx): horizontal stretch with scale factor 1/q; (x, y) → (x ÷ q, y)
- For y = pf(x) + b, do the stretch or reflection first and the vertical translation last
Common mistake
Students read f(x − 3) as a move to the left and f(2x) as a stretch with factor 2. Inside changes work in reverse: f(x − 3) moves right by 3 and f(2x) multiplies x-coordinates by ½.
Worked example
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