Question

use the graph of y = f(x) to graph the function g(x) = f(x - 4) - 3. choose the correct graph of g below. (graphs of y = f(x) and options a, b, c, d are shown)

Explanation:

Step1: Analyze Transformation Rules

The function \( g(x) = f(x - 4) - 3 \) involves two transformations: a horizontal shift and a vertical shift. For a function \( y = f(x - h) + k \), \( h \) is the horizontal shift (right if \( h>0 \)) and \( k \) is the vertical shift (down if \( k<0 \)). Here, \( h = 4 \) (shift right 4 units) and \( k=-3 \) (shift down 3 units).

Step2: Apply Horizontal Shift

Take key points from the graph of \( y = f(x) \). For example, if a vertex of \( f(x) \) is at \( (x_1, y_1) \), after shifting right 4 units, its \( x \)-coordinate becomes \( x_1 + 4 \), and \( y \)-coordinate remains \( y_1 \) for now.

Step3: Apply Vertical Shift

After the horizontal shift, shift each point down 3 units. So the new \( y \)-coordinate is \( y_1 - 3 \).

Step4: Identify Correct Graph

Compare the transformed points with the options. The graph that shows all key points of \( f(x) \) shifted right 4 and down 3 should be the correct one. Among the options, the graph that matches this transformation (shifting right 4, down 3) is the correct choice (e.g., if we analyze the positions, the graph with points moved appropriately as per the shifts).

Answer:

(Assuming the correct graph is, for example, Option D after analyzing shifts; but since the exact graph details are from the image, the correct graph would be the one where each point of \( f(x) \) is shifted 4 units right and 3 units down. If we assume the correct option is D, then) D. (Graph corresponding to \( g(x)=f(x - 4)-3 \) after shifts)