"Reflection through x axis" is one of the different types of transformations of 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 "Reflection through x-axis",
we have explained the rule that we apply to make reflection
transformation through x-axis of a function.

How to do reflection transformation through x-axis? (Rule)

Let y = f(x) be a function.

In the above function, if we want to do reflection through the x-axis, "y" has to be replaced by "-y" and we get the new function

-y = f(x)

-y = f(x) can also be written as y = -f(x)

The graph of y = -f(x) can be obtained by reflecting the graph of y = f(x) through the x-axis.

It can be done by using the rule given below.

Reflection through the x-axis - Example

Once
students understand the above mentioned rule which they have to apply
for reflection through x axis, they can easily do this kind of
transformations of functions.

Let us consider the following example to have better understanding of reflection through x axis.

Question :

Perform the following transformation to the function y = √x.

"a reflection through the x - axis"

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 reflection transformation through the x-axis, we have to replace "y" by "-y" in the given function y = √x.

Step 2 :

So, the formula that gives the requested transformation is

- y = √x or y = - √x

Step 3 :

The graph y = - √x can be obtained by reflecting the graph of y = - √x through the x-axis using the rule given below.

(x , y ) -----------------> ( x , -y )

Step 4 :

The graph of the original function (given function)

Step 5 :

The graph of the transformed function.

How to sketch the reflected graph through the x-axis ?

In the above function, we can easily sketch the reflected graph through the x-axis.

For some other functions, students may find it difficult to sketch the reflected graph.

For
example, students may find it difficult to sketch the reflected image
of the triangle whose vertices are A ( -2, 1), B (2, 4) and C
(4, 2)

To
get the reflected image through the x-axis, they just have to apply the
rule (x,y) ---------> (x, -y) in the above three vertices.

When they do so, they can get the vertices of the reflected image.

They are A' ( -2, -1) , B' ( 2, -4 ) and C' ( 4, -2)