1. **State the problem:** We need to analyze the function $y = -|x^2 - 2|$ and find ordered pairs (points) on its graph. 2. **Understand the function:** The function involves the absolute value of a quadratic expression $x^2 - 2$, then negated. The absolute value ensures the expression inside is always non-negative, and the negative sign outside flips the values to non-positive. 3. **Calculate some values:** Let's find $y$ for some values of $x$ to get ordered pairs. - For $x=0$: $$y = -|0^2 - 2| = -| -2| = -2$$ So the point is $(0, -2)$. - For $x=1$: $$y = -|1^2 - 2| = -|1 - 2| = -| -1| = -1$$ So the point is $(1, -1)$. - For $x=-1$: $$y = -|(-1)^2 - 2| = -|1 - 2| = -1$$ So the point is $(-1, -1)$. - For $x=2$: $$y = -|2^2 - 2| = -|4 - 2| = -2$$ So the point is $(2, -2)$. - For $x=-2$: $$y = -|(-2)^2 - 2| = -|4 - 2| = -2$$ So the point is $(-2, -2)$. 4. **Summary:** The function outputs negative values or zero (if $x^2 - 2 = 0$). The vertex points where $x^2 - 2 = 0$ are at $x = \\pm \sqrt{2} \approx \pm 1.414$, where $y=0$. 5. **Final ordered pairs:** Some points on the graph are: $$ (0, -2), (1, -1), (-1, -1), (2, -2), (-2, -2), (\sqrt{2}, 0), (-\sqrt{2}, 0) $$ These points can be plotted on the coordinate plane to visualize the graph of $y = -|x^2 - 2|$.