**Objectives:** In constructing graphs of functions, a very useful technique is to work with a collection of *standard functions* and then apply a variety of geometrical techniques to transform the graph of the standard function into the graph of the desired function. In this tutorial, we define the three
types of geometrical transformations and get the algebraic representation of
these transformations. Examples of these transformations are investigated.

After working through these materials, the student should be able

- to graph a function which is obtainable from a standard function by geometrical transformations;
- to algebraically express a function which is obtainable from a standard function by geometrical transformations;
- to recognize the graph of a function which is obtainable from a standard function by geometrical transformations.

**Modules:**

- We start with a
java applet to explore graphically the basic geometric transformations of functions.
We now proceed to express these geometrical transformations in algebraic terms:

**Translations of Functions.**Suppose that**y = f(x)**is a function and**c > 0**; then- the graph of
**y = f(x - c)**is obtained by translating the graph of**y = f(x)****c**units to the right; - the graph of
**y = f(x + c)**is obtained by translating the graph of**y = f(x)****c**units to the left; - the graph of
**y = f(x) + c**is obtained by translating the graph of**y = f(x)****c**units upwards; and - the graph of
**y = f(x) - c**is obtained by translating the graph of**y = f(x)****c**units downwards.

- the graph of
**Examples**.**LiveMath notebook**illustrating horizontal translations.**LiveMath notebook**illustrating vertical translations.**LiveMath notebook**illustrating general translations.

**Compression and Stretching of Functions.**Suppose that**y = f(x)**is a function and**c > 1**; then- the graph of
**y = f(cx)**is obtained by compressing horizontally the graph of**y = f(x)**by a factor of**c**units; - the graph of
**y = f(x/c)**is obtained by stretching horizontally the graph of**y = f(x)**by a factor of**c**units; - the graph of
**y = cf(x)**is obtained by stretching vertically the graph of**y = f(x)**by a factor of**c**units; - the graph of
**y = (1/c)f(x)**is obtained by compressing vertically the graph of**y = f(x)**by a factor of**c**units;

**Examples**.**LiveMath notebook**illustrating horizontal compressions and stretchings.**LiveMath notebook**illustrating vertical compressions and stretchings.

**Reflections.**Suppose that**y = f(x)**is a function; then- the graph of
**y = f(-x)**is obtained by reflecting the graph of**y = f(x)**across the**y**-axis; - the graph of
**y = -f(x)**is obtained by reflecting the graph of**y = f(x)**across the**x**-axis;

**Examples**.**LiveMath notebook**illustrating the reflection of a graph across the y-axis.**LiveMath notebook**illustrating the reflection of a graph across the x-axis.

**Quiz**on transformation of graphs.**Some More Examples**.- Given a function, draw the graphs of functions obtained by shifting the graph of a given function using
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator