What is a function transformation?

A change to a function’s rule that moves or reshapes its graph in a predictable way: shifting it, flipping it, or stretching it.

Why does x − 3 move the graph to the right?

The inside is the input. To get the same output as before, you must put in an x that is 3 bigger, so every point moves 3 to the right. Changes inside the brackets always act backwards.

Do transformations work on any function?

Yes. The same moves work on a line, a parabola, an absolute value graph, a square root, or any other function.

Part 21 of 30 · TransformationsSee the whole course

Function transformations: shifts, flips and stretches

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

Short answer

A transformation changes a graph by changing the rule in one place. Outside the brackets a change acts on the heights: lift, lower, flip, stretch. Inside the brackets a change acts on x, and it goes the other way: x − 3 slides the graph 3 to the right.

Start from a parent graph

Every transformation starts from a basic shape, the parent. The parabola y = x² is the easiest to watch, because it has a clear bottom point, the vertex. Other parents you will meet are |x| and √x. A transformation takes the parent and moves or reshapes it, without changing what kind of graph it is.

The four moves

Lift or lower: add a number at the end. y = x² + 2 is the parabola lifted by 2.
Slide sideways: change the inside. y = (x − 3)² slides 3 to the right, not left.
Flip: a minus in front flips it upside down. A minus on x flips it left to right.
Stretch or squash: a number in front. y = 2x² is pulled taller, so it looks narrower. y = ½x² is flatter.

The one thing to remember: outside acts on heights, inside acts on x and goes backwards. The reason is that the inside is what you feed in. To get the same output as before you must feed in an x that is 3 bigger, so the whole picture moves 3 to the right.

Worked example: y = −2(x − 1)² + 3

Read the moves from the rule, starting from y = x²:

Inside: x − 1, so slide 1 to the right.
In front: −2, so stretch by 2 and flip it over.
At the end: + 3, so lift it up 3.

The vertex moves from (0, 0) to (1, 3). Check with a point: the parent point (2, 4) lands on (3, −5), because −2(3 − 1)² + 3 = −2(4) + 3 = −5 ✓.

Play with it below. The gray parent stays put, and the orange dot follows the parent point (2, 4) to where it lands.

y = −2(x − 1)² + 3

Sideways h 1

Up and down k 3

Stretch in front 2

inside: x − 1, 1 to the right
in front: −2, stretched 2× and flipped
at the end: + 3, 3 up

The vertex of the bowl moves from (0, 0) to (1, 3). The parent point (2, 4) lands on (3, −5).

In what order?

Do the moves in the same order you would calculate the rule: the inside first, then the number in front, then the number at the end. For y = −2(x − 1)² + 3: put in x, take away 1, square, multiply by −2, then add 3.

Watch it explained in under a minute

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

The five mistakes to avoid

Sliding the wrong way. (x − 3) moves right and (x + 3) moves left.
Thinking 2x² is wider. A bigger number in front makes it narrower.
Mixing up the two flips. A minus in front flips over the x-axis, a minus on x flips over the y-axis.
Doing the moves in the wrong order. Shifts that use the inside come first, the number at the end comes last.
Confusing f(x) + 2 with f(x + 2). The first lifts the graph, the second slides it left.

Try it yourself

Describe the moves from y = x². Tap a question to see the answer.

1. y = x² − 4

Lower it by 4. The vertex moves to (0, −4).

2. y = (x + 3)²

Slide it 3 to the left. The vertex moves to (−3, 0).

3. y = −(x − 2)² + 5

Slide 2 right, flip it over, lift it 5. The vertex is (2, 5) and it opens downward like an arch.

Where this fits in

The next four posts take one move at a time, starting with vertical shifts.