Science Library - free educational site

Transformations of Functions

Functions can be translated (shifted vertically or hoizontally without change of scale or shape), reflected (resulting in a curve that is the mirror image of another curve, where the 'mirror' may be an axis or a line should as y = x), and magnified (sretched or shrunk in one or more directions).

Sine and cosine functions have transformations similar to linear and quadratic functions.

Transformations of Functions

Functions have transformations of the following types:

$f(x) + d$ Vertical translation Translation d units vertically upwards
$f(x + c)$ Horizontal translation Translation c units horizontally to the left
$-f(x)$ Reflection across x-axis Every value of y is made negative
$f(-x)$ Reflection across the y-axis Every x-value is made negative
$a⋅f(x)$ Vertical stretch Stretch by a factor of a in the vertical direction
$f(bx)$ Horizontal stretch Stretch by a factor of 1/b in the horizontal direction
Mathematics question

The function $f(x)$ can be translated upwards (d), translated sideways (-c), stretched vertically (a), stetched horizontally ($1/b$), reflected in the $x$-axis (-f(x)) and reflected in the $y$-axis (f(-x)), by means of different constants in the equation: $y= af(bx - c) + d$

Rational Function Transformations

A function of the form $x↦{ax+b}/{cx+d}$, where $x≠d/c$, and $a$, $b$, $c$ and $d$ are constants, is called a simple rational function.

These functions have both a horizontal and vertical asymptote.

In general, the function $y = {ax + b}/{cx + d}$ can be rewritten to $y= A/{B(x-h)} + k$

where A is the vertical stretch factor, B is the reciprocal of the horizontal stretch factor, and $( \table h;k )$ is the translation in the $( \table x;y )$ directions.

Transformation of a function

Multiple Transformation Example

Find the function $y=g(x)$ which results when transforming the function $x↦1/x$ by: a vertical stretch by factor $1/2$, a horizontal stretch by factor $3$, and a translation of $[\table 2;-3]$:

The vertical stretch of $1/2$ [general: ${f(x)}/2$] results in $1/{2x}$

The horizontal stretch of $3$ [general: ${f(x/3)}$] results in $3/{2x}$

The translation of of $[\table 2;-3]$ [general: ${f(x-2)-3}$] results in $3/{2(x-2)}-3$

$g(x) = 3/{2(x-2)}-3 = 3/{2x-4}-{3(2x-4)}/{(2x-4)} = {3-6x+12}/{(2x-4)} = {-3x+{15}/2}/{(x-2)}$

The asymptotes of $1/x$ are $x=0$ and $y=0$, so the asymptotes of the transformation are the translation parameters: $x-2=0$ ⇒ $x=2$, and $y=a/c={-3}/1=-3$

Invariant Points

As the name suggests, invariant points do not move under a transformation. If there is no translation, a polynomial which passes through zero will still pass through zero irrespective of any stretching or inversions which occur.

Mathematics question

The zeros of $f(x)$ become vertical asymptote values of $1/{f(x)}$

The vertical asymptote values of $f(x)$ become zeros of $1/{f(x)}$

Maximum values of $f(x)$ become minimum values of $1/{f(x)}$

Minimum values of $f(x)$ become maximum values of $1/{f(x)}$

When $f(x)>0$, $1/{f(x)}>0$

When $f(x)<0$, $1/{f(x)}<0$

When $f(x)→0$, $1/{f(x)}→±∞$

When $1/{f(x)}→0$, $f(x)→±∞$

Content © All rights reserved. Created : September 26, 2014

Latest Item on Science Library:

The most recent article is:


View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week


Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on


Great Scientists

Robert Brown

1773 - 1858

Robert Brown was a pioneer of the use of the microscope for botanical and cell research. He discovered the phenomenon of Brownian Motion, the erratic movement of pollen grains in water, which inspired Albert Einstein to predict the discovery of atoms in a 1905 paper.

Robert Brown
Transalpine traduzioni

Quote of the day...

Capitalism is the astounding belief that the wickedest of men will do the wickedest of things for the general good of everyone.

ZumGuy Internet Promotions