Function Transformation Math Example 2

Solution

1. 1
   Rewrite: $g(x) = -\sqrt{2(x+2)}$. Identify parameters: $a=-1$, $b=2$, $h=-2$, $k=0$.
2. 2
   Step 1 — Horizontal compression by $\frac{1}{2}$ (due to $b=2$): $(x,y) \to (x/2, y)$. Key point $(4,2)\to(2,2)$.
3. 3
   Step 2 — Horizontal shift left $2$ (due to $h=-2$): $(x,y)\to(x-2,y)$. Key point $(2,2)\to(0,2)$. Step 3 — Reflect over $x$-axis (due to $a=-1$): $(x,y)\to(x,-y)$. Key point $(0,2)\to(0,-2)$. Domain: $2x+4\geq0 \Rightarrow x\geq-2$.