A translation slides a graph without turning or resizing it.
Adding a number outside the function, y = f(x) + a, moves the graph up by a. Changing x inside the brackets, y = f(x − a), moves the graph right by a. The shape stays the same, and every point moves by the same amount.
This lesson builds on interpreting gradient and intercept in graphs and transformations. Check the Cambridge syllabus page for which transformation notation your tier uses.
How does a translation change the equation?
There are two separate moves, and each one is independent.
Vertical move: y = f(x) + a. Each y-value grows by a, so the graph rises by a (or falls if a is negative). This one matches your intuition.
Horizontal move: y = f(x − a). The graph moves right by a. The sign feels backwards because the new graph needs a larger x to reach the same output. If f(x) had a vertex at x = 0, then f(x − 3) has its vertex where x − 3 = 0, so at x = 3.
Combined, y = f(x − a) + b is a translation by the column vector (a, b).
Worked example
Start with y = x². Write the equation after a translation by the vector (3, −2), and find the new positions of four points.
Step 1, equation: moving 3 right replaces x with (x − 3). Moving 2 down subtracts 2. So y = (x − 3)² − 2.
Step 2, move the points: add 3 to each x and subtract 2 from each y.
| Original | New point |
|---|---|
| (0, 0) | (3, −2) |
| (1, 1) | (4, −1) |
| (2, 4) | (5, 2) |
| (−1, 1) | (2, −1) |
Step 3, check with the equation: at x = 4, (4 − 3)² − 2 = 1 − 2 = −1. At x = 5, (5 − 3)² − 2 = 4 − 2 = 2. At x = 2, (2 − 3)² − 2 = 1 − 2 = −1. All match the table.
The vertex of y = x² was (0, 0). It is now at (3, −2), which you can read straight from the equation.
The mistake to watch for
A common slip is to move the graph the wrong way for a bracket change.
Mistaken answer: “y = (x + 2)² is y = x² shifted 2 to the right.”
The student saw a plus and moved in the positive direction.
Test one point to correct it. For y = (x + 2)², the vertex is where x + 2 = 0, so x = −2.
The graph moved 2 to the left. Substitute x = −2 and you get y = 0, which confirms the vertex is at (−2, 0).
Check yourself
Try these without a calculator, then open each answer.
1. Describe the translation that takes y = x² to y = x² + 5, and state the new vertex.
Show answer
The 5 is added outside, so the graph moves 5 units up, vector (0, 5). The vertex moves from (0, 0) to (0, 5).
2. Describe the translation that takes y = x² to y = (x + 4)².
Show answer
The vertex is where x + 4 = 0, so x = −4. The graph moves 4 units left, vector (−4, 0).
3. The line y = 2x is translated by the vector (3, 0). Find the new equation and check it.
Show answer
Replace x with (x − 3): y = 2(x − 3) = y = 2x − 6. Check: the point (0, 0) moves to (3, 0), and 2 × 3 − 6 = 0, so (3, 0) lies on the new line.
Where this leads next
Next, see how a flip works in reflecting a graph, then mix both ideas in the graphs and transformations practice set. The quadratic structure explorer lets you change a quadratic and watch the vertex move, and the graph model explorer shows how a fitted line or curve depends on its parameters.
Some students follow a worked translation but hesitate on the sign when a new function appears. In online one-to-one Mathematics tuition, a teacher can give you fresh functions until testing one point becomes automatic.