Question

describe the transformation of f(x) to g(x).

f(x) = sin x

a. f(x) is shifted down frac(π,2) units to g(x).
b. f(x) is shifted up frac(π,2) units to g(x).
c. f(x) is shifted up 2 units to g(x).
d. f(x) is shifted up 1 unit to g(x).

Explanation:

Step1: Analyze the original function

The original function is \( f(x)=\sin x \), which has a midline (the horizontal line that the graph oscillates around) at \( y = 0 \) (since the sine function oscillates between - 1 and 1, and the mid - point of - 1 and 1 is 0).

Step2: Analyze the transformed function \( g(x) \)

Looking at the graph of \( g(x) \), we can see that its midline is at \( y = 1 \). To find the vertical shift, we compare the midline of \( f(x) \) and \( g(x) \). The midline of \( f(x) \) is \( y = 0 \) and the midline of \( g(x) \) is \( y=1 \). A vertical shift of a function \( y = f(x) \) to \( y=f(x)+k \) represents a shift up by \( k \) units when \( k>0 \). Here, since the midline has moved from \( y = 0 \) to \( y = 1 \), the function \( f(x)=\sin x \) is shifted up by 1 unit to get \( g(x) \).

We can also check the y - intercept. For \( f(x)=\sin x \), when \( x = 0 \), \( f(0)=\sin(0)=0 \). For \( g(x) \), when \( x = 0 \), from the graph, \( g(0) = 1 \). The difference in the y - intercepts (0 for \( f(x) \) and 1 for \( g(x) \)) also indicates a vertical shift of 1 unit up.

Answer:

D. \( f(x) \) is shifted up 1 unit to \( g(x) \)