Question - Understanding Transformation of Functions
Solution:
The function f(x) = x^2 is given, and the function g(x) is defined as g(x) = 4f(x) - 4 = 4(x^2) - 4.The transformation from f(x) to g(x) involves two steps:1. The term 4f(x) indicates that the function f(x) is multiplied by 4, which stretches or scales the graph vertically by a factor of 4. This means that for any x, the value of g(x) will be four times the value of f(x), but the shape of the graph remains the same (it will just be stretched).2. The term -4 at the end of the expression indicates that the graph is then shifted down by 4 units. It is a vertical translation downwards.Putting these together, we get:A vertical stretch of the graph of f(x) by a factor of 4, followed by a downward shift of the graph by 4 units.The correct answer to the question in the image is D: The graph of g(x) is the graph of f(x) shifted down 4 units. The option about stretching is incorrect because the vertical multiplication by 4 does not stretch the graph of f(x) by a factor of 1/4; it stretches it by a factor of 4. The combination of these transformations makes D the best choice among the provided options.
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