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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
Graph
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 $1/{f(x)}→0$, $f(x)→±∞$

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