Question

s the graph of $f(x)=sqrt{x}$. form it to make it as the graph of $g(x)=sqrt{x + 1}-3$.

Explanation:

Step1: Identify horizontal shift

For $g(x)=\sqrt{x+1}-3$, compare to $f(x)=\sqrt{x}$. The $x+1$ inside the square root means shift left 1 unit.

Step2: Identify vertical shift

The $-3$ outside the square root means shift down 3 units.

Step3: Transform key points

• Vertex of $f(x)$: $(0,0)$. Shift left 1, down 3: $(0-1, 0-3)=(-1,-3)$.
• Another point on $f(x)$: $(1,1)$. Shift left 1, down 3: $(1-1,1-3)=(0,-2)$.
• Another point on $f(x)$: $(4,2)$. Shift left 1, down 3: $(4-1,2-3)=(3,-1)$.

Answer:

The graph of $g(x)=\sqrt{x+1}-3$ is the graph of $f(x)=\sqrt{x}$ shifted 1 unit to the left and 3 units down. Its key points include $(-1,-3)$, $(0,-2)$, and $(3,-1)$, and it follows the same curved shape as the original square root graph, starting at $(-1,-3)$ and increasing slowly as $x$ increases.