Question - Understanding Function Transformations through Graphs
Solution:
The image presents a math problem with two graphs. The graph of a function f(x) is shown in gray, and the graph of another function g(x) is shown in pink. We are informed that g(x) has the same shape as f(x), which implies that g(x) is a transformation of f(x).The transformation appears to involve both a horizontal shift to the right and a vertical shift downward. Specifically, the point (0,1) on f(x) has been mapped to the point (3,0) on g(x). This indicates a horizontal shift of 3 units to the right and a vertical shift of 1 unit down.Now, let's examine the provided options to determine the correct equation for g(x):A. $$ g(x) = f(x − 3)^2 − 1 $$: Incorrect because this suggests f(x) is squared after a shift, which is inconsistent with uniform scaling.B. $$ g(x) = f(x − 1)^2 − 3 $$: Incorrect because the inner subtraction should reflect a horizontal shift of 3 units, not 1.C. $$ g(x) = f(x − 3) − 3 $$: Almost correct, but it shows an extra vertical shift down by 3 units instead of 1 unit.D. $$ g(x) = f(x − 3) − 1 $$: Correct because this represents a rightward shift of 3 units (x − 3) and a downward shift of 1 unit (− 1).Therefore, the correct answer is:D. $$ g(x) = f(x − 3) − 1 $$
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