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 reading a graph with intercepts 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 can follow a worked translation but hesitate on the sign when a new function appears. That is the kind of pattern a teacher will find quickly in online one-to-one Mathematics tuition.