Question

use the graph of ( y = f(x) ) to graph the function ( g(x)=\frac{1}{2} f(x - 3) ).
choose the correct graph of ( g ) below.

Explanation:

Step1: Horizontal shift

For the function \(g(x)=\frac{1}{2}f(x - 3)\), compared to \(y = f(x)\), the graph of \(y=f(x)\) is shifted 3 units to the right. Because for a function \(y = f(x - h)\), when \(h>0\), the graph shifts \(h\) units to the right.

Step2: Vertical compression

The factor \(\frac{1}{2}\) in \(g(x)=\frac{1}{2}f(x - 3)\) causes a vertical compression. For a function \(y = af(x)\) where \(0 < a<1\), the graph of \(y = f(x)\) is vertically compressed by a factor of \(a\).

Looking at the options:

• Option A: The graph is shifted 3 units to the right and vertically compressed.
• Option B: There is no horizontal shift (should be shifted 3 units to the right)
• Option C: The vertical compression factor is not \(\frac{1}{2}\) (the y - values are not halved in the correct way)
• Option D: There is no horizontal shift (should be shifted 3 units to the right)

Answer:

A.