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Given \[y=-2f\left( 1-2x \right)+3\]. How do you describe the transformation?

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Last updated date: 20th Jun 2024
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Answer
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Hint: Assume \[f\left( x \right)\] as the reference function. Now, compare the given function \[y=-2f\left( 1-2x \right)+3\] with the general form \[y=af\left( b\left( x+c \right) \right)+d\]. Find the corresponding values of a, b, c and d. Using these values find the vertical compression or stretch (a), horizontal compression or stretch (b), horizontal shift of the function (c) and vertical shift of the function (d).

Complete step-by-step solution:
Here, we have been provided with the function \[y=-2f\left( 1-2x \right)+3\] and we are asked to explain the transformations of the function. Generally, we assume \[y=f\left( x \right)\] as our reference function.
Now, the general form of the transform of a function is given as \[y=af\left( b\left( x+c \right) \right)+d\]. Here, a, b, c and d have their own meanings. For this general function the transform is described as: -
1. If \[\left| a \right|>1\] then vertical stretch takes place and if 0 < a < 1 then vertical compression takes place.
2. If ‘a’ is negative then the function is reflected about the x – axis.
3. If \[\left| b \right|>1\] then horizontal stretch takes place and if 0 < b < 1 then horizontal compression takes place.
4. If ‘b’ is negative then the function is reflected about y – axis.
5. If ‘c’ is negative then the function is shifted \[\left| c \right|\] units to the right and if ‘c’ is positive then the function is shifted \[\left| c \right|\] units to the left.
6. If ‘d’ is negative then the function is shifted \[\left| k \right|\] units down and if ‘d’ is positive then the function is shifted \[\left| k \right|\] units up.
Now, let us write our given function \[y=-2f\left( 1-2x \right)+3\] in the general form and compare the values of a, b, c and d. So, we have,
\[y=-2f\left( -2\left( x-\dfrac{1}{2} \right) \right)+3\]
On comparing the values of a, b, c and d, we have,
\[\Rightarrow \] a = -2, b = -2, \[c=\dfrac{-1}{2}\] and d = +3
So, the description of the transform of the function can be given as: -
1. The function will have a vertical stretch of 2 units and it will be reflected about x – axis.
2. The function will have a horizontal stretch of 2 units and it will be reflected about y – axis.
3. The function will have a horizontal shift of \[\dfrac{1}{2}\] units to the right.
4. The function will have a vertical shift of 3 units in the upward direction.

Note: One may note that you can also understand the situations by taking the examples of some trigonometric functions like: - sine or cosine function. Assume \[y=\sin x\] and transform it into the function \[y=A\sin \left( B\left( x+C \right) \right)+D\]. Draw the graph of the two functions and compare the phase shifts, vertical and horizontal shifts etc. You must remember the six rules that we have stated in the solution. These transform rules are very useful in drawing the graph of functions which are provided to us in the chapter ‘area under curve’ in the topic ‘Integration’.