Transformations of functions (across all function families)
Transformations of Functions
Transformations of Functions is a topic in Functions in the Common Core State Standards. It is aligned to Standard F-BF.3, which requires students to build new functions by transforming a parent function.
A transformation shifts, stretches, or reflects the graph of a parent function — \(y=a\,f(b(x-c))+d\) — without changing its underlying family.
Theory
A transformation moves or reshapes the graph of a parent function without changing its basic family. Starting from \(y=f(x)\):
- \(f(x)+d\) — vertical shift (up if \(d>0\), down if \(d<0\)).
- \(f(x-c)\) — horizontal shift (right if \(c>0\), left if \(c<0\)) — note it moves opposite to the sign inside.
- \(a\,f(x)\) — vertical stretch (\(|a|>1\)) or compression (\(|a|<1\)); reflects across the \(x\)-axis if \(a<0\).
- \(f(bx)\) — horizontal compression (\(|b|>1\)) or stretch (\(|b|<1\)) by factor \(\dfrac{1}{|b|}\); reflects across the \(y\)-axis if \(b<0\).
The transformation of \(y=f(x)\):
with \(a\) vertical scale/reflection, \(b\) horizontal scale/reflection, \(c\) horizontal shift, and \(d\) vertical shift.
How to apply transformations
- Identify the parent function \(f\).
- Handle horizontal changes (inside \(f\)): factor out \(b\), read the shift \(c\), remember it acts oppositely.
- Handle vertical changes (outside \(f\)): the multiplier \(a\) and the added constant \(d\).
- Track a key point (like the vertex) through each step to check your work.
Read the transformations from the outside in, remembering horizontal changes are “inside” and act oppositely:
- \(x-3\): shift right 3
- \(2\,f(\ldots)\): vertical stretch by factor 2
- \(+1\): shift up 1
So: right 3, stretch vertically by 2, up 1.
Match each piece to the parent \(x^2\):
| \((x+2)^2\) | \(\Rightarrow\) | shift left 2 |
| \(-4\) | \(\Rightarrow\) | shift down 4 |
The vertex moves from \((0,0)\) to \((-2,-4)\).
The negative sign's position decides the axis of reflection.
- \(y=-\sqrt{x}\): the minus is outside, so it negates outputs — reflect across the \(x\)-axis (curve turns downward).
- \(y=\sqrt{-x}\): the minus is inside, so it negates inputs — reflect across the \(y\)-axis (curve opens to the left).
A factor \(b\) inside, \(f(bx)\), scales horizontally by \(\dfrac{1}{b}\) — the opposite of what it looks like.
| \(f(2x)\) | \(\Rightarrow\) | horizontal compression by \(\dfrac{1}{2}\) |
Every point is pulled toward the \(y\)-axis to half its \(x\)-distance.
Common pitfalls
Frequently asked questions
How do you shift a graph up or down?
Add a constant outside the function: \(f(x)+d\) shifts up by \(d\) (down if \(d\) is negative).
Why does f(x-3) shift right instead of left?
Because the input reaches the same value later: to make \(x-3\) equal the old input, \(x\) must be 3 larger, so the whole graph slides right 3.
What is the difference between -f(x) and f(-x)?
\(-f(x)\) reflects across the \(x\)-axis (negates outputs); \(f(-x)\) reflects across the \(y\)-axis (negates inputs).
Does f(2x) stretch or compress the graph?
It compresses horizontally by a factor of one half. A factor \(b\) inside scales the graph by \(1/b\) horizontally.