Translations

What is a Translation?

A translation shifts a graph without changing its shape or size.

Think of sliding a piece of paper across a table.

Vertical Shifts

$f(x) + k$

Add a constant outside the function to shift vertically.

• $k > 0$ → shift up
• $k < 0$ → shift down

Example: Start with $f(x) = x^2$

• $f(x) + 3 = x^2 + 3$ → shift up 3
• $f(x) - 2 = x^2 - 2$ → shift down 2

Every point moves up or down by the same amount.

Horizontal Shifts

$f(x - h)$

Subtract a constant inside the function to shift horizontally.

• $h > 0$ → shift right
• $h < 0$ → shift left

Warning: This is backwards from what you might expect!

$f(x - 3)$ shifts right 3, not left.

$f(x + 2)$ shifts left 2, not right.

Why is it backwards?

Think about when the output equals zero.

For $f(x) = x^2$, we get $f(0) = 0$ at $x = 0$.

For $g(x) = (x - 3)^2$, we get $g(3) = 0$ at $x = 3$.

The vertex moved from $x = 0$ to $x = 3$. That’s a shift right.

Example: Start with $f(x) = x^2$

• $f(x - 4) = (x - 4)^2$ → shift right 4
• $f(x + 1) = (x + 1)^2$ → shift left 1

Combining Shifts

You can shift both horizontally and vertically:

$f(x - h) + k$

Example: $g(x) = (x - 2)^2 + 3$

This takes $f(x) = x^2$ and:

• Shifts right 2
• Shifts up 3

The vertex moves from $(0, 0)$ to $(2, 3)$.

Quick Reference