Question

1. $y \leq \sqrt{x + 3} - 4$

Explanation:

Step1: Identify the parent function

The parent function is $y=\sqrt{x}$, which has a domain of $x\geq0$ and range of $y\geq0$.

Step2: Apply horizontal shift

The term $x+3$ shifts the parent function left 3 units. So the domain becomes $x+3\geq0 \implies x\geq-3$.

Step3: Apply vertical shift

The $-4$ shifts the function down 4 units, so the range becomes $y\geq-4$.

Step4: Graph the boundary curve

Plot $y=\sqrt{x+3}-4$ as a solid curve (since the inequality is $\leq$, the boundary is included).

Step5: Determine the shaded region

Test a point below the curve, e.g., $x=-3, y=-5$:
$\boldsymbol{-5 \leq \sqrt{-3+3}-4 \implies -5 \leq -4}$, which is true. So shade the region below and including the curve $y=\sqrt{x+3}-4$, with the curve defined only for $x\geq-3$.

Answer:

The graph is a solid curve of $y=\sqrt{x+3}-4$ (starting at $(-3,-4)$, increasing slowly) with all area below and including this curve shaded, and the curve exists only for $x\geq-3$.