Question

what kind of transformation converts the graph of $f(x) = 4x - 8$ into the graph of $g(x) = x - 2$?
options: horizontal shrink, vertical shrink, vertical stretch, horizontal stretch

Explanation:

Step1: Recall transformation rules

For linear functions \( y = mx + b \), a vertical transformation of \( y = f(x) \) is \( y = a\cdot f(x) \), where \( |a| < 1 \) is a vertical shrink, \( |a| > 1 \) is a vertical stretch. A horizontal transformation is \( y = f(bx) \), where \( |b| < 1 \) is a horizontal stretch, \( |b| > 1 \) is a horizontal shrink.

Step2: Express \( g(x) \) in terms of \( f(x) \)

Given \( f(x)=4x - 8 \), factor out 4: \( f(x)=4(x - 2) \). Now, \( g(x)=x - 2 \). Notice that \( g(x)=\frac{1}{4}f(x) \), because \( \frac{1}{4}f(x)=\frac{1}{4}(4x - 8)=x - 2 \).

Step3: Identify the transformation type

Since \( g(x)=\frac{1}{4}f(x) \), and \( |\frac{1}{4}| < 1 \), this is a vertical shrink (multiplying the function by a factor between 0 and 1 vertically shrinks the graph).

Answer:

B. vertical shrink