Final answer:
The function g(x) = (0.5x)^2 represents a horizontal compression of the original function f(x) = x^2 by a factor of 2. This transformation results in the graph being squeezed toward the y-axis compared to the original graph.
Step-by-step explanation:
When considering the transformation of the function f(x) = x2 into the new function g(x) = (0.5x)2, we notice a change in the rule for g(x) that affects the graph of the original function. This operation involves squaring the product of 0.5 and x. What actually happens here is a horizontal compression of the graph of f(x) by a factor of 2. We can see this by comparing the input values of f(x) and g(x).
If we choose a certain value for x in f(x), such as x = 4, then f(4) = 42 = 16. For the function g(x), if we also plug in 4 for x, we get g(4) = (0.5×4)2 = 22 = 4. The result for g(4) is the same as f(2), since 22 = 4. This illustrates how g(x) compresses the original function horizontally because g(x) reached the value of 4 at x = 4, whereas f(x) reached it at x = 2, hence the horizontal compression by a factor of 2.
Transformations such as these where a factor is applied to the variable x inside the function result in horizontal stretching or compression, depending on whether the factor is greater or less than 1, respectively. Since in our case the factor is 0.5 or 1/2, this leads to a compression. The graphical representation would show that every point on the original graph of f(x) has moved horizontally closer to the y-axis by a factor of 2, to form the new graph of g(x).