# 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

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.

### 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.

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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