Question

use transformations of ( f(x)=sqrt{x} ) to graph the following function. ( g(x)=4sqrt{x + 3}+5 ) choose the correct graph below

Explanation:

Step1: Identify horizontal shift

For $g(x)=4\sqrt{x+3}+5$, compare to $f(x)=\sqrt{x}$. The $x+3$ inside the square root shifts the graph of $f(x)$ left 3 units. The domain of $g(x)$ is $x+3\geq0 \implies x\geq-3$.

Step2: Identify vertical stretch

The coefficient 4 outside the square root vertically stretches the graph of $\sqrt{x+3}$ by a factor of 4.

Step3: Identify vertical shift

The $+5$ outside the square root shifts the stretched graph up 5 units.

Step4: Find key reference point

The starting point of $f(x)=\sqrt{x}$ is $(0,0)$. Applying transformations:

1. Left 3: $(0-3, 0)=(-3,0)$
2. Vertical stretch by 4: $(-3, 0\times4)=(-3,0)$
3. Up 5: $(-3, 0+5)=(-3,5)$

Another point: For $f(1)=1$. Transform:

1. Left 3: $1-3=-2$
2. Stretch by 4: $1\times4=4$
3. Up 5: $4+5=9$, so point $(-2,9)$

Answer:

A. <Graph with starting point at (-3,5), increasing curve matching the transformed square root shape>