Identify the value of k that would correctly characterize the horizontal shift in the graph if the line on the left represents y = f(x) and the line to the right represents y = f(x -k)

Question

Identify the value of k that would correctly characterize the horizontal shift in the graph if the line on the left represents y = f(x) and the line to the right represents y = f(x -k)
Left has values of (15,45), (20,40), (25,35)
Right values (5,45),(10,40), (15,35)

GPT-4o mini · Answer

To find the value of $ k $ that characterizes the horizontal shift in the graph of $ y = f(x) $ to the graph of $ y = f(x - k) $, we need to analyze the given points on both graphs. The original graph points (on the left) are: - $ (15, 45) $ - $ (20, 40) $ - $ (25, 35) $ The shifted graph points (on the right) are: - $ (5, 45) $ - $ (10, 40) $ - $ (15, 35) $ The transformation from $ y = f(x) $ to $ y = f(x - k) $ involves a horizontal shift by $ k $ units to the right. Thus, a point $ (a, b) $ from $ y = f(x) $ will move to $ (a + k, b) $ in $ y = f(x - k) $. Now, let's compare points: 1. The point $ (15, 45) $ shifts to $ (5, 45) $: $$ 15 - k = 5 $$ $$ k = 15 - 5 = 10 $$ 2. The point $ (20, 40) $ shifts to $ (10, 40) $: $$ 20 - k = 10 $$ $$ k = 20 - 10 = 10 $$ 3. The point $ (25, 35) $ shifts to $ (15, 35) $: $$ 25 - k = 15 $$ $$ k = 25 - 15 = 10 $$ In all cases, we find that $ k = 10 $. Thus, the value of $ k $ that correctly characterizes the horizontal shift is $ \boxed{10} $.