Question

analyzing a relationship
which equation describes the relationship of the
graphs of f(x) and g(x)?
○ g(x) = f(x) + 1
○ g(x) = f(x) − 1
○ g(x) = f(x + 1)
× g(x) = f(x − 1)

Explanation:

Step1: Identify Vertical Shift

Check vertical positions. The blue graph (g(x)) is above the orange (f(x)) by 1 unit. So vertical shift: \( g(x) = f(x) + 1 \) or check horizontal shift. Wait, horizontal shift: if \( g(x) = f(x + 1) \), it's left shift. Wait, let's take a point. At x=0, f(0)=0? Wait, orange at x=0 is 0? Wait blue at x=0 is 1. Wait, no: blue graph at x=0 is 1, orange at x=0 is 0? Wait no, looking at the graph: blue (upper) and orange (lower). Let's check the y-intercept. Blue has y-intercept 1, orange has y-intercept 0? Wait no, orange at x=0 is 0? Wait, no, the orange graph passes through (0,0)? Wait blue passes through (0,1), orange through (0,0)? Wait, no, the blue graph is above orange. So vertical shift: g(x) = f(x) + 1? Wait no, wait the options: first option \( g(x) = f(x) + 1 \), second \( f(x)-1 \), third \( f(x+1) \), fourth \( f(x-1) \). Wait, maybe horizontal shift. Wait, let's take a point on f(x) and see where it is on g(x). Suppose f(x) is the orange graph, g(x) is blue. Let's take x=0: f(0)=0, g(0)=1. If \( g(x) = f(x) + 1 \), then g(0)=f(0)+1=0+1=1, which matches. Wait, but earlier thought was wrong. Wait, maybe the blue is g(x), orange is f(x). So g(x) = f(x) + 1? But the red cross is on \( g(x)=f(x-1) \), which is wrong. Let's re-examine. Wait, the graph: blue and orange. Let's check the horizontal shift. If we have a horizontal shift, for example, \( g(x) = f(x + 1) \) means shift left by 1, \( f(x - 1) \) shift right by 1. Wait, take a point on f(x): say f(1) is some value, g(0) should be f(1) if \( g(x)=f(x+1) \). Wait, maybe the correct answer is \( g(x) = f(x) + 1 \)? Wait no, let's check the y-values. At x=0, g(x) (blue) is 1, f(x) (orange) is 0. So g(0) = f(0) + 1. At x=1, g(1) is 2, f(1) is 1. So g(1) = f(1) + 1. So the vertical shift is up by 1, so \( g(x) = f(x) + 1 \). But the red cross was on \( f(x-1) \), which is wrong. So the correct answer should be \( g(x) = f(x) + 1 \). Wait, but let's confirm. So the first option is \( g(x) = f(x) + 1 \), which would mean adding 1 to each y-value of f(x) to get g(x), which matches the graph (blue is above orange by 1 unit vertically).

Step2: Confirm the Correct Option

Check the vertical shift: g(x) is f(x) shifted up by 1 unit, so \( g(x) = f(x) + 1 \). The other options: \( f(x)-1 \) would shift down, \( f(x+1) \) left, \( f(x-1) \) right. The graph shows vertical shift up, so first option is correct.

Answer:

A. \( g(x) = f(x) + 1 \)