Question

suppose g(x) is a transformation of f(x) such that f(x) translates left 8 units, stretches vertically by a factor of \\(\frac{1}{4}\\) and translates down 12 units. enter the function g(x).

Explanation:

Step1: Determine the base function

Assume the base function \( f(x)=\sqrt{x} \) (since the original graph resembles the square - root function).

Step2: Apply the horizontal translation

The rule for a horizontal translation of a function \( y = f(x) \) to \( y=f(x + h) \) (where \( h>0 \) shifts left). If \( f(x) \) is translated left 8 units, the function becomes \( f(x + 8)=\sqrt{x + 8} \).

Step3: Apply the vertical stretch

The rule for a vertical stretch of a function \( y = f(x) \) by a factor \( a \) is \( y=a\cdot f(x) \). Here \( a=-\frac{1}{4} \), so the function becomes \( -\frac{1}{4}\sqrt{x + 8} \).

Step4: Apply the vertical translation

The rule for a vertical translation of a function \( y = f(x) \) down \( k \) units is \( y=f(x)-k \). Here \( k = 12 \), so the function \( g(x)=-\frac{1}{4}\sqrt{x + 8}-12 \).

Answer:

\( g(x)=-\frac{1}{4}\sqrt{x + 8}-12 \)