Function Transformation Calculator
This function transformation calculator names every transformation that takes a parent function to the one you were given, in the order they apply.
Type -2|x - 1| + 3 and it tells you: start from |x|, shift 1 right, stretch vertically by 2, reflect in the x-axis, shift 3 up.
Everything Comes From Four Numbers
Function transformations all fit one shape, however different the questions look:
y = a · f(b(x − h)) + k
f is the parent — the plain shape underneath. The four numbers do the rest, and they split cleanly in two. a and k sit outside and act on the output. b and h sit inside and act on the input.
That split is the whole topic. The two outside behave the way they read; the two inside behave backwards.
| Number | Where | What it does |
|---|---|---|
| k | outside | shifts up (or down if negative) |
| a | outside | stretches vertically; a negative flips it top to bottom |
| h | inside | shifts right when it is subtracted |
| b | inside | squashes horizontally when it is bigger than 1 |
The One That Costs Marks
Here is √(2x − 6). How far does it shift?
Six, says almost everybody. It is three.
The 2 has to come out of the bracket first: 2x − 6 = 2(x − 3). Only once b is outside does the number next to x mean a shift, and by then it is 3, not 6. Every step after this one inherits the error, so a wrong answer here is wrong all the way down.
This page factors the inside before it reads anything, and says so when it has had to.
Why Inside Behaves Backwards
It is the one part of this topic worth understanding rather than memorising, and it takes a sentence.
y = f(x − 3) asks for the value f had at x − 3. So at x = 5 you get what f had at 2 — the graph is showing you old news, three units late. Late means further right. Subtracting shifts right.
The same argument covers b. y = f(2x) reaches whatever f did at 10 by the time x is only 5, so everything arrives twice as early and the picture is squashed to half its width. Multiplying by 2 halves the graph rather than doubling it.
The Order Is Part of the Answer
Naming the transformations without their order is half a mark at best.
Work from the inside out. Anything done to x happens first: the horizontal squash, then the reflection in the y-axis if b is negative, then the horizontal shift. Anything done to the whole function happens after: the vertical stretch, the reflection in the x-axis if a is negative, then the vertical shift last of all.
Swap the shift and the stretch on the same axis and you get a different graph. That is why "shift 3 right and stretch by 2" and "stretch by 2 and shift 3 right" are not the same instruction.
The Six Shapes Underneath
Almost every question in this topic is built on one of six parents.
| Parent | What it looks like |
|---|---|
| x | a straight line through the origin |
| |x| | a V with its corner at the origin |
| x² | a curve with one turning point at the origin |
| x³ | an S through the origin |
| √x | half a curve on its side, starting at the origin |
| 1/x | two branches in opposite quadrants |
Naming the parent first is worth doing even when it seems obvious, because everything after it is measured against that shape.
Sketching Without Plotting
You do not need to work out a table of values. Take the points you already know on the parent and put each one through the same transformations.
On |x| the corner sits at (0, 0), with (−1, 1) and (1, 1) beside it. Put those three through −2|x − 1| + 3 and they land at (1, 3), (0, 1) and (2, 1). Three points and you have the V: corner up at (1, 3), opening downwards, twice as steep as before.
That is faster than any table, and it is what the calculator above prints alongside the answer. If you want the picture itself rather than the points, a graphing tool is the right thing — this page is for saying what happened.
Where You Have Already Met This
You have been doing transformations for longer than you think.
Writing a quadratic in the form a(x − h)² + k is exactly this: h and k are the two shifts and a is the vertical stretch, which is why that form tells you the turning point at a glance. The vertex calculator handles that case properly, and completing the square is how you get there from the expanded form.
Graph shifts and reflections turn up on their own too: a function is even when reflecting in the y-axis changes nothing, and odd when reflecting in both axes changes nothing, which the even and odd function calculator tests directly.
How to Use This Function Transformation Calculator
Type the function as it is written: -2|x - 1| + 3, or sqrt(2x - 6), or 3/(x - 1). A y = in front is ignored.
Coefficients may be fractions or decimals, and the inside is factored for you before anything is read off. Shifting a domain rather than a graph is a different question, and the domain and range calculator covers it; combining two functions rather than reshaping one is handled by operations on functions.
Function Transformation Calculator FAQ
What is a parent function?
Why does x − 3 shift right and not left?
How far does √(2x − 6) shift?
Does the order of transformations matter?
What does a negative in front do?
Why does multiplying x by 2 squash the graph?
Can I do transformations without drawing?
Which shapes does this page read?
Find the parent, factor the inside, then read the four numbers off and apply them from the inside out. The shift is the one to check twice.
