Question

select the correct answer. in which direction must the graph of f(x) = x be shifted to produce the graph of g(x) = f(x) - 4? a. left and down b. right and up c. down d. up

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x)+k \), if \( k>0 \), the graph shifts up; if \( k < 0 \), the graph shifts down. Here, \( g(x)=f(x)-4 \), so \( k=-4 \) (negative).

Step2: Analyze Horizontal Shift (None Here)

The transformation \( g(x)=f(x)-4 \) has no horizontal shift (no change to the \( x \)-term inside \( f \)), only a vertical shift. Since \( k = - 4<0 \), the vertical shift is down. There's no horizontal shift (left/right) as the \( x \) in \( f(x) \) remains unchanged. So the graph of \( f(x) \) is shifted down to get \( g(x) \). Wait, wait, no—wait, the options: Wait, maybe I misread? Wait, the function is \( g(x)=f(x)-4 \), which is a vertical shift down. But let's check the options again. Wait, the original function is \( f(x)=x \), and \( g(x)=f(x)-4=x - 4 \). The transformation from \( y = x \) to \( y=x - 4 \) is a vertical shift down (since we subtract 4 from the function value, which moves each point \( (x,y) \) to \( (x,y - 4) \), so down). But wait, the options: A is left and down, B right and up, C down, D up. Wait, maybe I made a mistake. Wait, no—horizontal shift: for \( y = f(x - h) \), shift right if \( h>0 \), left if \( h < 0 \). Here, \( g(x)=f(x)-4 \), so \( h = 0 \) (no horizontal shift), \( k=-4 \) (vertical shift down). So the direction is down? But wait, the options: Wait, the problem says "in which direction must the graph of \( f(x)=x \) be shifted to produce \( g(x)=f(x)-4 \)". So vertical shift down. So option C? Wait, but let's re - check. The transformation \( g(x)=f(x)-4 \) is a vertical translation. The rule is: \( y = f(x)+k \), \( k>0 \) up, \( k < 0 \) down. So \( k=-4 \), so down. There's no horizontal shift (left/right) because there's no \( (x - h) \) or \( (x+h) \) inside \( f \). So the shift is down (vertical), no horizontal shift. So the correct option is C.

Answer:

C. down