Translations of Quadratic Functions

Q: How do I remember which number is horizontal and which is vertical?

Inside the parentheses touches the $x$, so it moves the graph in the $x$ direction (horizontally). Outside the parentheses is added to the whole output $y$, so it moves the graph in the $y$ direction (vertically).

Every parabola in this lesson starts from the parent function $f(x) = x^{2}$ . A translation slides that graph to a new spot without changing its U-shape at all — same width, same steepness, new location.

There are only two moves: vertical (up or down) and horizontal (left or right). The number outside the square controls the vertical move and behaves exactly the way you expect. The number inside the square controls the horizontal move — and its sign flips, which is where every mistake in this topic comes from.

Vertical shifts: outside the square

Adding a constant after the squared term changes the output: $y = x^{2} + k$ . Every point on the parabola moves up by $k$ if $k$ is positive, or down if $k$ is negative. So $y = x^{2} + 5$ is the parent graph shifted up $5$ units, and $y = x^{2} - 3$ is shifted down $3$ units.

The outside number keeps its sign: $+5$ really means up $5$ .

Horizontal shifts: inside the square

Adding a constant inside the parentheses changes the input: $y = (x - h)^{2}$ . This slides the graph horizontally — and the direction is the opposite of the sign you see. $y = (x - 2)^{2}$ moves right $2$ units; $y = (x + 4)^{2}$ moves left $4$ units.

Here is why. The parent graph has its vertex where the squared quantity equals zero. For $(x - 2)^{2}$ , that happens at $x = 2$ , so the vertex lands at $x = 2$ — to the right. The inside sign flips; the outside sign stays the same.

Both shifts at once

Put the two moves together and you get $y = (x - h)^{2} + k$ : right or left by $h$ , up or down by $k$ , so the vertex lands at $(h, k)$ .

Below, the black parabola is the parent $y = x^{2}$ and the blue parabola is $y = (x - 2)^{2} + 3$ — the same shape moved right $2$ and up $3$ , with its vertex at $(2, 3)$ .

Worked examples

Example 1: describe the translation

How does $y = (x - 2)^{2} + 3$ compare to $y = x^{2}$ ?

Inside number — flip the sign, so the graph moves right

x - 2 \;\Rightarrow\; \text{right } 2

Outside number — keeps its sign, so the graph moves up

+3 \;\Rightarrow\; \text{up } 3

The vertex lands at

(2, 3)

Answer: Shifted right $2$ units and up $3$ units.

Example 2: a left-and-down shift

How does $y = (x + 4)^{2} - 1$ compare to $y = x^{2}$ ?

Rewrite the inside as a subtraction to see

h

x + 4 = x - (-4)

So the graph moves left

4

units

h = -4

The outside

-1

moves it down

1

unit

k = -1

Answer: Shifted left $4$ units and down $1$ unit; vertex $(-4, -1)$ .

Example 3: write the equation

Write the equation of $y = x^{2}$ shifted left $1$ unit and up $4$ units.

Left

1

means the inside is

x + 1

— the sign flips

(x + 1)^{2}

4

means add

4

outside

(x + 1)^{2} + 4

Answer: $y = (x + 1)^{2} + 4$

Try one yourself

Common questions

Why does the inside sign flip?

The vertex sits where the squared quantity equals zero. For $(x - 2)^{2}$ that is $x = 2$ , so writing minus $2$ pushes the vertex to positive $2$ — the direction opposite the sign you see. The outside number acts after the squaring, so it keeps its sign.

Does a translation change the shape of the parabola?

No. A translation only slides the graph. It stays exactly as wide and opens the same direction. Changing the shape takes a dilation — a number multiplying the squared term.

How do I remember which number is horizontal and which is vertical?

Inside the parentheses touches the $x$ , so it moves the graph in the $x$ direction (horizontally). Outside the parentheses is added to the whole output $y$ , so it moves the graph in the $y$ direction (vertically).

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