About the topic "Horizontal translations of functions"

"Horizontal translations of functions" is one of the different types of transformations in functions.

Even though students can get this stuff on internet, they do not understand exactly what has been explained.

To make the students to understand the stuff "Horizontal translation of a function", we have explained the rule that we apply to make horizontal translation of a function.

Horizontal Translation - definition

A horizontal translation "slides" an object a fixed
distance either on the right side or left side. The original object and its translation
have the same shape and size, and they
face in the same direction.

In simple words, horizontal translation means, it just moves the given figure either on the right or left without rotating, re-sizing or anything else.

How to do horizontal translation of a function? (Rule)

Let y = f(x) be a function and "k" be a constant.

In the above function, if "x" is replaced by "x-k" , we get the new function y = f(x-k).

The graph of y= f(x-k) can be obtained by the translating the graph of y = f(x) to the right by "k" units if "k" is a positive number.

In case "k" is a negative number, the graph of y = f(x) will be translated to the left by |k| units.

Moreover, if the the point (x,y) is on the graph of y = f(x), then the point (x+k , y) is on the graph y = f(x-k).

For example, if k =3, the graph of y = f(x) will be translated to the right by "3" units.

If k = -3, the graph of y = f(x) will be translated to the left by "3" units.

Horizontal Translation - Example

Once students understand the above mentioned rule which they have to apply for horizontal translation, they can easily make horizontal translations of functions.

Let us consider the following example to have better understanding of horizontal translation of a function.

Question :

Perform the following transformation to the function

y = √x.

"a translation to the right by 3 units"

And also write the formula that gives the requested transformation and draw the graph of both the given function and the transformed function

Solution:

Step 1 :

Since we do a translation to the right by "3" units, we have to replace "x" by "x-3" in the given function y = √x.

Step 2 :

So, the formula that gives the requested transformation is y = √(x-3)

Step 3 :

The graph y = √(x-3) can be obtained by translating the graph of y = √x to the right by "3" units.

Step 4 :

The graph of the original function (given function)