About the topic "Vertical translations of functions"

"Vertical 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 "Vertical translation of a function", we have explained the rule that we apply to make vertical translation of a function.

Vertical Translation - definition

A vertical translation "slides" an object a fixed
distance either up or down. The original object and its translation
have the same shape and size, and they
face in the same direction.

In simple words, vertical translation means, it just moves the given figure either up or down without rotating, re sizing or anything else.

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

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

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

The graph of y= f(x) + k can be obtained by translating the graph of y = f(x) towards upward by "k" units.

In case, "y" is replaced by "y + k" , we get the new function

y + k = f(x) or y = f(x) - k.

The graph of y= f(x) - k can be obtained by translating the graph of y = f(x) towards downward by "k" units.

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

Vertical Translation - Example

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

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

Question :

Perform the following transformation to the function

y = √x.

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

Step 2 :

So, the formula that gives the requested transformation is

y -3 = √x or y = √x + 3

Step 3 :

The graph y = √x + 3 can be obtained by translating the graph of y = √x toward upward by "3" units.

Step 4 :

The graph of the original function (given function)