describe how to graph the solution of
$y \le -x^2 + 2x$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Quadratic Functions", "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ] } ## Identify the boundary curve and inequality type $$ \begin{aligned} &\text{Boundary Equation: } y = -x^2 + 2x\ &\text{Inequality Symbol: } \le \implies \text{Solid boundary line, shade below} \end{aligned} $$ ## Determine key features of the parabola $$ \begin{aligned} &\text{Vertex: } x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1 \implies y = -(1)^2 + 2(1) = 1 \implies (1, 1)\ &\text{y-intercept: } (0, 0)\ &\text{x-intercepts: } -x(x - 2) = 0 \implies (0, 0) \text{ and } (2, 0)\ &\text{Direction: } a = -1 < 0 \implies \text{Opens downward} \end{aligned} $$ ## Determine the shaded region using a test point $$ \begin{aligned} &\text{Test Point: } (1, 0)\ &0 \le -(1)^2 + 2(1) \implies 0 \le 1 \quad (\text{True})\ &\implies \text{Shade the region containing } (1, 0) \text{ (below/inside the parabola)} \end{aligned} $$ To graph the solution of $y \le -x^2 + 2x$: 1. **Graph the boundary parabola** $y = -x^2 + 2x$ as a **solid line** because the inequality includes "or equal to" ($\le$). The parabola opens downward with its vertex at $(1, 1)$ and x-intercepts at $(0, 0)$ and $(2, 0)$. 2. **Shade the region below** (inside) the parabola. You can verify this by testing the point $(1, 0)$: since $0 \le -(1)^2 + 2(1)$ is true ($0 \le 1$), shade the region containing this point. { "elements": [ { "type": "inequality", "params": [ { "js": "-x*x + 2*x", "latex": "-x^2 + 2x" }, [1, -2] ], "properties": { "inverse": false, "strict": false, "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "point", "params": [[1, 1]], "properties": { "name": "Vertex (1,1)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[0, 0]], "properties": { "name": "(0,0)", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "(2,0)", "size": 3, "color": "#583C87", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities in Two Variables" ] }

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Identify the boundary curve and inequality type

Determine key features of the parabola

Determine the shaded region using a test point

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