Question

below is the graph of $f(x)=\sqrt{x}$.
transform it to make it as the graph of $g(x)=\sqrt{x + 2}-4$.

Explanation:

Step1: Analyze horizontal shift

For the function \( f(x)=\sqrt{x} \), to get \( \sqrt{x + 2} \), we use the horizontal shift rule. The transformation \( f(x + h) \) shifts the graph of \( f(x) \) left by \( h \) units when \( h>0 \). Here, \( h = 2 \), so we shift the graph of \( f(x)=\sqrt{x} \) 2 units to the left.

Step2: Analyze vertical shift

For the function \( \sqrt{x + 2} \), to get \( \sqrt{x + 2}-4 \), we use the vertical shift rule. The transformation \( f(x)-k \) shifts the graph of \( f(x) \) down by \( k \) units when \( k>0 \). Here, \( k = 4 \), so we shift the graph of \( \sqrt{x + 2} \) (after horizontal shift) 4 units down.

Answer:

First, shift the graph of \( f(x)=\sqrt{x} \) 2 units to the left, then shift the resulting graph 4 units down to obtain the graph of \( g(x)=\sqrt{x + 2}-4 \).