Intersection of a parabolic region and a rhombus

Text

Consider the two sets $A = \{(x;y)\mid y \le -x^2 + 4\}, \qquad B = \{(x;y)\mid |x| + |y| \le 3\}.$

• (a) represent $A$ and $B$ separately, highlighting the boundary;
• (b) represent $A \cap B$;
• (c) decide whether $P(0;3)$, $Q(-2;0)$, $R(1;-3)$ belong to $A\cap B$.

Solution

(a) $A$ is the region below the parabola $y=-x^2+4$ (vertex $(0;4)$, $x$-intercepts $\pm2$), solid boundary. $B$ is the rhombus with vertices $(\pm3;0)$ and $(0;\pm3)$, solid boundary.

(b) $A\cap B$ is the rhombus $B$ minus the part lying above the parabola (a small upper cap).

(c) Check both conditions:

• $P(0;3)$: $A$? $3\le4$ ✓; $B$? $3\le3$ ✓ → $P\in A\cap B$.
• $Q(-2;0)$: $A$? $0\le0$ ✓; $B$? $2\le3$ ✓ → $Q\in A\cap B$.
• $R(1;-3)$: $A$? $-3\le3$ ✓; $B$? $4\le3$? No → $R\notin A\cap B$.

$\boxed{P\in A\cap B,\quad Q\in A\cap B,\quad R\notin A\cap B}$