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.

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: $y= af(bx - c) + d$

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.

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 $y=a/c={-3}/1=-3$

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 $1/{f(x)}→0$, $f(x)→±∞$

Content © Renewable-Media.com. All rights reserved. Created : September 26, 2014

The most recent article is:

View this item in the topic:

and many more articles in the subject:

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 ScienceLibrary.info.

1825 - 1895

Thomas Henry Huxley, 1825 - 1895, was an English biologist, most famous for his fervent defence of Darwin's Theory of Evolution.

Website © renewable-media.com | Designed by: Andrew Bone