f(x) is vertically shrunk by a factor of [tex]$\frac{1}{3}$[/tex]. How will you represent [tex]$f(x)$[/tex] after transformation?
A. [tex]$3 f(x)$[/tex]
B. [tex]$f(\frac{1}{3} x)$[/tex]
C. [tex]$f(3 x)$[/tex]
D. [tex]$\frac{1}{3} f(x)$[/tex]
Answer (2)
A vertical shrink by a factor of 3 1 means multiplying the function by 3 1 .
The transformed function is represented as 3 1 f ( x ) .
This scales down all the output values of the function.
The final answer is 3 1 f ( x ) .
Explanation
Understanding the Problem We are given a function f ( x ) that undergoes a vertical shrink by a factor of 3 1 . Our goal is to determine the representation of the transformed function.
Applying the Vertical Shrink To vertically shrink a function f ( x ) by a factor of k , we multiply the function by k . In this case, the function is vertically shrunk by a factor of 3 1 , so k = 3 1 .
Representing the Transformed Function Therefore, the transformed function is 3 1 f ( x ) . This means that for every value of x , the new function's value is 3 1 of the original function's value.
Final Answer The correct representation of the function f ( x ) after a vertical shrink by a factor of 3 1 is 3 1 f ( x ) .
Examples
Imagine you have a recipe that makes a cake, represented by the function f ( x ) , where x is the amount of ingredients. If you want to make a smaller cake that's one-third the size of the original, you would multiply the entire recipe by 3 1 . This is exactly what a vertical shrink does to a function – it scales down all the output values by a certain factor, just like scaling down a recipe. Understanding function transformations helps in various real-life scenarios, such as adjusting quantities, scaling designs, or modifying models.
The function f ( x ) after being vertically shrunk by a factor of 3 1 is represented as 3 1 f ( x ) . The correct answer is option D. This transformation reduces all output values of the function by that factor, making it appear flatter on the graph.
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