Translating functions
Also known as: function shifts, translation of functions
Translating a function means shifting its graph horizontally or vertically without changing its shape. Adding k outside the function, f(x) + k, shifts the graph up k units, while replacing x with x − h, f(x − h), shifts it right h units.
A translation slides the graph of a function up, down, left, or right without rotating, stretching, or reflecting it. Every point on the graph moves the same distance in the same direction, so the shape stays identical — only the position changes.
Vertical translations come from adding or subtracting outside the function: g(x) = f(x) + k shifts the graph up k units when k > 0 and down when k < 0. Horizontal translations come from changes inside the function: g(x) = f(x − h) shifts the graph right h units, and g(x) = f(x + h) shifts it left. For example, if f(x) = x², then f(x − 3) + 2 = (x − 3)² + 2 moves the parabola 3 units right and 2 units up, placing its vertex at (3, 2).
The horizontal direction is the classic trap: the shift runs opposite to the sign inside the parentheses. Subtracting inside, f(x − 3), moves the graph right, not left. One way to see why: f(x − 3) needs an input 3 units larger to produce the same output f gave before, so every feature of the graph lands 3 units later on the x-axis. Vertical shifts behave intuitively because they act directly on the output.
The SAT's Advanced Math questions test function translations regularly — identifying the equation of a shifted graph, predicting where a vertex or intercept moves, or combining horizontal and vertical shifts in a single transformation.
Key takeaways
- A translation shifts a graph without changing its shape or orientation.
- f(x) + k shifts the graph up k units; f(x) − k shifts it down.
- f(x − h) shifts the graph right h units; f(x + h) shifts it left — opposite to the sign inside.
- Changes outside the function move the graph vertically; changes inside move it horizontally.
- The SAT tests translations through shifted equations, moved vertices, and combined shifts.