ACT® · Math: Functions — Composition, Transformation, and Inverse Functions
Translation by Specified Horizontal and Vertical Amounts (AF 706)
Translating a graph means sliding it without changing its shape. Changes inside the parentheses move the graph horizontally and backwards from what you'd expect; changes outside move it vertically in the expected direction.
A translation slides every point on a graph the same distance in the same direction, leaving the shape untouched. Vertical shifts are the intuitive ones: adding a number outside the function, like in y = f(x) + k, pushes the whole graph up by k, and subtracting pushes it down. Your eye and the algebra agree here.
Horizontal shifts are the ones that trip people up, because they happen inside the parentheses and run backwards. In y = f(x - h), every output value that used to show up at x now shows up at x + h, since you need a bigger input to cancel out the subtraction and recreate the same number f was getting before. So the whole picture moves right, even though there's a minus sign. Same logic in reverse for y = f(x + h): plugging in a smaller x now produces what used to require the larger x + h, so the graph slides left by h.
This idea is the quiet partner of composition and stretching: a translation is really just f composed with a simple shift function, and it combines with vertical stretches or flips to build the full transformed equation of a parent graph like y = x2 or y = abs(x). Once shifts feel automatic, reading off a vertex or intercept after several transformations stacked together becomes just careful bookkeeping instead of a new skill.
Key terms
- Translation
- A slide of a graph up, down, left, or right that keeps its size and shape unchanged.
- Horizontal shift
- A move left or right caused by replacing x with x - h or x + h inside the function.
- Vertical shift
- A move up or down caused by adding or subtracting a constant outside the function.
- Parent graph
- The basic, untransformed version of a function family, such as y = x2 or y = abs(x).
- Vertex
- The turning point of a graph like a parabola or absolute value V, used to track shifts easily.
Watch it explained
- How it worksHow does replacing x with x - h slide the graph right instead of left?
- Worked exampleFind the equation of y = |x| after moving it 3 units right and 2 units down.
- Worked exampleFind the new vertex of y = (x + 1)2 - 4 after a shift of 5 right and 6 up.
- Common mistakeWhy doesn't y = f(x + 2) move the picture to the right?
- Going furtherWhat happens to a horizontal shift when the input inside the function is 2x rather than x?
Check yourself
End of topic.