Question

use transformations of ( f(x)=sqrt4{x} ) to graph the following function. ( g(x)=4sqrt4{x - 3}+1 ) choose the correct graph below. oa. image ob. image oc. image od. image

Explanation:

Step1: Identify horizontal shift

For $g(x)=4\sqrt{x-3}+1$, compare to $f(x)=\sqrt{x}$. The $x-3$ inside the square root shifts $f(x)$ right 3 units. Domain of $g(x)$: $x-3\geq0 \implies x\geq3$.

Step2: Identify vertical stretch

The coefficient 4 outside $\sqrt{x-3}$ vertically stretches $f(x)$ by factor 4. For $x=3$, $g(3)=4\sqrt{0}+1=1$. For $x=4$, $g(4)=4\sqrt{1}+1=5$.

Step3: Identify vertical shift

The $+1$ outside shifts the stretched/shifted graph up 1 unit. This confirms the y-intercept (at x=3) is 1, and the graph increases steeply due to the vertical stretch.

Step4: Match to correct graph

The correct graph has a starting point at $(3,1)$, increases steeply, and lies in the region $x\geq3$, $y\geq1$.

Answer:

(Select the option corresponding to the graph with starting point $(3,1)$, steep upward curve for $x\geq3$; based on standard transformations, this matches the graph that is shifted right 3, stretched vertically by 4, and shifted up 1 from $f(x)=\sqrt{x}$)