Why does f(x − 3) move the graph to the right?

The graph reaches the same height when the inside is the same as before. x − 3 equals 0 when x = 3, so every point arrives 3 units later, to the right.

What does f(x + h) do to a graph?

It slides the graph h units to the left. The inside x + h is zero when x = −h, so the bottom or start of the graph moves to x = −h.

Does a horizontal shift change the domain and range?

It can change the domain, because the allowed inputs move with the graph. For √(x + 4) the domain becomes x ≥ −4. The range stays the same.

How is a horizontal shift different from a vertical shift?

A horizontal shift changes the input, as in f(x − h), and moves the graph left or right. A vertical shift adds a number outside, as in f(x) + k, and moves it up or down.

Part 23 of 30 · TransformationsSee the whole course

Horizontal shifts: why f(x − h) goes the other way

By The Pip team · October 24, 2026 · 7 min read

Short answer

Changing the inside, f(x − h), slides the graph h units to the right. f(x + h) slides it left. It goes the opposite way from what the sign suggests, because the graph reaches its bottom where the inside equals zero, and x − 3 equals 0 at x = 3.

The inside is the input

A function takes an input and gives an output. In f(x − 3), the thing being fed into f is x − 3, not x. To get the output f(0), you need x − 3 = 0, so you need x = 3. The graph reaches the same height three steps later: it has slid 3 to the right.

See it in a table

Take f(x) = x² and use f(x − 3) = (x − 3)²:

x: 1, 2, 3, 4, 5
x − 3: −2, −1, 0, 1, 2
(x − 3)²: 4, 1, 0, 1, 4

The bottom of the bowl, where the output is 0, is now at x = 3. The old bottom was at x = 0, so the whole graph moved 3 to the right. Slide h in the figure below and check where the gold dot lands. It is always above the x that makes the inside zero.

y = (x − 3)²

Number taken from x, h 3

inside: x − 3 = 0 when x = 3
the bottom moves from x = 0 to x = 3
a point at x = 2 moves to x = 5

Every point moves 3 to the right, even though the sign inside is a minus. The bottom sits where the inside equals 0: x = 3.

Three examples

y = (x + 2)²: the inside x + 2 is zero at x = −2, so the bottom is at (−2, 0): left 2.
y = |x − 2|: the corner is at (2, 0): right 2.
y = √(x + 4): it starts at (−4, 0), so its domain is x ≥ −4: left 4.

What changes, and what does not

The domain can change. √x needs x ≥ 0, but √(x + 4) needs x ≥ −4. See domain and range.
The range does not change. The heights are the same, only moved sideways.
The shape does not change.

Both shifts together

y = (x − 3)² + 1 is x² slid 3 to the right and lifted 1: the vertex is at (3, 1). The inside moves x, the number at the end moves the height. Compare with vertical shifts.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The four mistakes to avoid

Reading the sign the wrong way. (x − 3) moves right and (x + 3) moves left.
Forgetting the domain moves too. For √(x + 4), x can no longer be below −4.
Mixing up f(x + 2) with f(x) + 2. The first slides it, the second lifts it.
Moving by the wrong amount when x has a multiplier. (2x − 6)² is 4(x − 3)², so it shifts 3, not 6. Factor out the number on x first.

Try it yourself

Tap a question to see the answer.

1. Where is the vertex of y = (x − 5)²?

The inside is zero at x = 5, so the vertex is (5, 0): the parabola slid 5 to the right.

2. Where is the corner of y = |x + 1| − 2?

The inside x + 1 is zero at x = −1, and the −2 lowers it. The corner is (−1, −2).

3. What is the starting point and domain of y = √(x − 2)? Does it pass through (6, 2)?

It starts at (2, 0), so the domain is x ≥ 2. At x = 6, √4 = 2, so yes, it passes through (6, 2).

Where this fits in

You can now slide a graph in both directions. Next, turning it over: reflections over the axes.