Question

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

choose the correct graph of ( g ) below.

options: a, b, c, d (with corresponding graphs)

Explanation:

Step1: Analyze the transformation

The function \(g(x)=\frac{1}{2}f(x - 2)\) involves two transformations: a horizontal shift and a vertical compression.

• Horizontal shift: The \(x-2\) inside the function \(f\) means a shift to the right by 2 units.
• Vertical compression: The \(\frac{1}{2}\) in front of \(f\) means a vertical compression by a factor of \(\frac{1}{2}\).

Step2: Apply the transformations to key points

Assume some key points on \(y = f(x)\). For example, if \((x_1,y_1)\) is a point on \(y = f(x)\), then the corresponding point on \(y = g(x)\) is \((x_1+ 2,\frac{1}{2}y_1)\).

Step3: Compare with the options

• Option A: Does not show the correct vertical compression.
• Option B: Shows the correct horizontal shift (right by 2) and vertical compression (by \(\frac{1}{2}\)).
• Option C: Incorrect transformation (appears to be a reflection or wrong type of transformation).
• Option D: Incorrect transformation (wrong combination of shifts and compressions).

Answer:

B