Translating a parabola

Translate the parabola $y = x^2$ by $2$ in the $x$-direction and $1$ in the $y$-direction. Replacing $x$ with $x - 2$ and adding $1$ gives $y = (x - 2)^2 + 1$.

Why moving right means subtracting

The puzzling part is that moving $+2$ to the right requires subtracting $2$ inside the formula. Let $(x, y)$ be a point on the translated graph. It came from a point of the original graph moved $2$ right and $1$ up, so the original point was $(x - 2,\ y - 1)$. That point lies on $y = x^2$.

Subtracting inside the formula is just undoing the move: it takes the new coordinates back to the old ones.

Vertex form

In general, shifting $y = x^2$ by $p$ horizontally and $q$ vertically yields $y = (x - p)^2 + q$, moving the vertex from the origin to $(p, q)$. This is called the vertex form, and its virtue is that the vertex can be read straight off. The horizontal and vertical shifts are independent, and applying them in either order gives the same result.

Shape is unchanged

A translation does not change the shape of the parabola. The coefficient of $x^2$ is still $1$, so the opening and the orientation are untouched; only the position differs, and the figure is congruent to the one we started with.

Any quadratic in general form can be put into vertex form by completing the square, which reveals exactly how far $y = x^2$ has been moved, and when the leading coefficient is not $1$, how far it has been stretched vertically as well.

The same rule for every function

The rule is not peculiar to parabolas. For any function, $y = f(x - p) + q$ is the graph shifted $p$ right and $q$ up.

• Circles
• Sine curves
• The V of an absolute value

All of them translate by exactly the same recipe. Here the vertex moves from the origin to $(2, 1)$; the large dots mark the two vertices.