Graph transformations
Translating up and down (Higher tier only)
A slides a graph without changing its shape.
y = stands for any graph, such as y = x². y = f(x) + a moves it up a (down if a is negative): every y-value goes up by a.
Example: the graph of y = f(x) has a turning point at (2, −3). Find the turning point of y = f(x) + 5.
- Adding 5 moves the graph up 5.
- x stays the same: 2. y goes up 5: −3 + 5 = 2.
- (2, 2)
Key terms
Translating left and right (Higher tier only)
y = f(x + a) moves the graph of y = f(x) left a. So y = f(x − 2) moves it right 2: curve C is y = (x − 2)².
Example: the graph of y = f(x) has a turning point at (2, −3). Find the turning point of y = f(x + 4).
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- 4 inside the bracket moves the graph left 4.
- x goes down 4: 2 − 4 = −2. y stays the same.
- (−2, −3)
Common mistake
Inside the bracket, the direction is the opposite of what you'd expect: f(x + 4) moves left, and f(x − 2) moves right.
Reflections (Higher tier only)
- reflects the graph in the : every y-value changes sign.
- y = f(−x) reflects the graph in the : every x-value changes sign.
Curve A is y = x² + 2x. Curve B is y = f(−x) = (−x)² + 2(−x) = x² − 2x: its reflection in the y-axis.
Example: the graph of y = f(x) has a turning point at (2, −3). Find the turning points of y = −f(x) and y = f(−x).
- −f(x): the y-coordinate changes sign: (2, 3)
- f(−x): the x-coordinate changes sign: (−2, −3)