Horizontal Translations of Functions :
In function graphing, a horizontal translation is a transformation that shifts the original graph either to the right side or left side by some units.
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.
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.
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.
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
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)
Step 4 :
The graph of the transformed function.
After having gone through the stuff given above, we hope that the students would have understood, how to do horizontal translation of a function.
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