Question

analyze the graphs of f(x) and g(x). write an equation for g(x) in terms of f(x). sample problem g(x) = -f(x) + 2 each point on f(x) has been reflected across the x-axis and translated up 2 units. enter the answer in the space provided. use numbers instead of words. g(x) = \boxed{}

Explanation:

Step1: Identify Transformations

First, check vertical shift and reflection. For \( f(x) \) and \( g(x) \), observe key points. Let's take a point on \( f(x) \), say \( (2,8) \) (from sample? Wait, no, the right graphs: let's analyze the given graphs. Wait, the left graph of \( f(x) \) and right \( g(x) \). Wait, maybe horizontal shift? Wait, no, let's check the sample. Wait, in the problem, we need to find \( g(x) \) in terms of \( f(x) \). Let's see the graphs: the first graph (left) has \( f(x) \) passing through (0,0), (1,1), (2,8). The second graph (right) for \( g(x) \): let's find its relation. Wait, maybe vertical shift and horizontal shift? Wait, no, let's check the y-intercept. Wait, maybe \( g(x) = f(x - 3) \)? Wait, no, let's check the points. Wait, the left graph: \( f(x) \) at x=0 is 0, x=1 is 1, x=2 is 8. The right graph: \( g(x) \) at x=3 is 0 (since it crosses x-axis at 3), x=4 is 1, x=5 is 8? Wait, no, the right graph's curve: when x=3, y=0; x=4, y=1; x=5, y=8? Wait, that would mean \( g(x) = f(x - 3) \). Wait, because shifting f(x) 3 units to the right. Let's verify: if \( f(x) \) has a point (0,0), then \( f(x - 3) \) would have (3,0), which matches the right graph's x-intercept at 3. Similarly, (1,1) in f(x) becomes (4,1) in g(x), and (2,8) becomes (5,8). So the transformation is horizontal shift 3 units right. So \( g(x) = f(x - 3) \).

Step2: Confirm the Transformation

Check the key points: \( f(0)=0 \), so \( g(3)=f(3 - 3)=f(0)=0 \), which matches. \( f(1)=1 \), so \( g(4)=f(4 - 3)=f(1)=1 \), matches. \( f(2)=8 \), so \( g(5)=f(5 - 3)=f(2)=8 \), matches. So the equation is \( g(x) = f(x - 3) \).

Answer:

\( g(x) = f(x - 3) \)