Math Revision 15.7 - Transforming Graphs

In these revision notes for Transforming Graphs, we cover the following key points:

• What is a parent function?
• What is a graph transformation?
• What are graph translations?
• How many translations are there?
• What are the conditions for each type of translation?
• What are the rules applied in horizontal translations of graphs?
• How to make combined translations of graphs?
• The same for vertical translations.
• What are reflections in graphs?
• How many reflections are there?
• What are the conditions for each type of reflection?
• How to make combined reflections of graphs?

Transforming Graphs Revision Notes

The parent function is the simplest from a list of functions of the same category.

There are several methods we can use to transform a parent graph, to obtain the graph of a new function of the same family. Some of them include:

Translations

A translation in mathematics is an action that allows carrying an object (shape, graph, figure, etc.) across the coordinate plane, keeping its size, direction and shape unchanged. The orientation and area of the original mathematical object remain therefore unchanged after a translation as well. Simply put, a translation in math is a vertical shift, horizontal shift, or a combination of the two. Below, we will deal with these two types of translations. Despite

The key feature of translations is that it involves only displacements of a graph from the original position but no changes in size or orientation.

There are two types of graphs translations:

a. Horizontal Translations

The first type of translation discussed in this tutorial is horizontal translation. In other words, in this section, we will see how a graph can shift due left or right to the original position. The following rules are applied in horizontal translations of graphs.

1. A horizontal translation by a units to the right of the graph of a function f(x) is described mathematically by f(x - m).
2. A horizontal translation by a units to the left of the graph of a function f(x) is described mathematically by f(x + m).

b. Vertical Translations

The second type of translation discussed in this tutorial is vertical translation. In other words, in this section, we will see how a graph can shift due up or down with respect to the original position.

The following rules are applied in horizontal translations of graphs.

1. A vertical translation by n units up of the graph of a function f(x) is described mathematically by f(x) + n.
2. A vertical translation by c units down of the graph of a function f(x) is described mathematically by f(x) - n.

When the coefficient a of a quadratic function is different from 1, we can factorize it and express the function in the form y = a(x² + bx/a + c/a). Again, the translation rules are applied for the part of the expression in brackets, and then the values obtained are multiplied by a if necessary.

If there is more than one basic translation to do in a graph, we do it one at a time, not simultaneously. For example, first, we can do the horizontal translation, then the vertical one. The reverse action is also valid.

Reflections

A reflection in mathematics involves a change in the position and orientation of an object but no change in size. Like in translations, our focus in this tutorial will be only on graphs' reflections, not on other mathematical objects. Again, we will deal only with the horizontal and vertical reflections. In the popular language, reflections are also known as 'flips'.

In more scientific terms, a reflection is a rigid transformation, which means that the size and shape of the figure do not change; the figures are congruent (have the same size and shape) before and after the transformation.

It is worth pointing out the fact that all reflections are made with respect to an axis of symmetry, as stated in previous tutorials. We will take the two axes of the coordinate system as symmetry axes in graphs' reflections.

There are two types of graphs reflection:

a. Horizontal reflections

The first type of graphs reflection is the horizontal reflection. It occurs when a graph flips horizontally in respect to a vertical axis called a symmetry axis.

Mathematically, this type of transformation occurs when the function y = f(x) becomes y = f(-x). This means we have a horizontal reflection with respect to the vertical axis when f(x) → f(x). This occurs when we substitute the variable x with -x in the function's formula. The new function obtained represents therefore the reflection of the original function in the horizontal direction.

b. Vertical reflections

The second type of graphs reflection is the vertical reflection. It occurs when a graph flips vertically with respect to a horizontal axis called a symmetry axis.

Mathematically, this type of transformation occurs when the function y = f(x) becomes y = -f(x). This means we have a vertical reflection with respect to the vertical axis when f(x) → -f(x). This occurs when we place a negative sign before the function's formula. The new function obtained represents therefore the reflection of the original function in the vertical direction.

There are other types of graph transformations besides those discussed in this tutorial. For example, a graph may increase (enlargement) or reduce (diminution) in size. We will discuss this transformation when dealing with vectors.

Another type of graphs' transformation is the rotation at a given angle with respect to the original graph. This graph transformation too will be explained in future chapters, after dealing with trigonometry.