which quadratic inequality does the graph below represent?

$y \ge x^2 - 3$
$y \le x^2 + 3$
$y \le x^2 - 3$
$y \ge x^2 + 3$

Question

which quadratic inequality does the graph below represent?

$y \ge x^2 - 3$
$y \le x^2 + 3$
$y \le x^2 - 3$
$y \ge x^2 + 3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Quadratic Graphs", "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ], "new_concepts": [], "current_concepts": [ "Quadratic Graphs", "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ] } ## Identify the boundary parabola equation $$ \begin{aligned} &\text{Vertex: } (0, -3)\ &\text{Point on parabola: } (2, 1)\ &y = a(x - h)^2 + k \implies 1 = a(2 - 0)^2 - 3 \implies 4 = 4a \implies a = 1\ &\text{Boundary equation: } y = x^2 - 3 \end{aligned} $$ ## Determine the inequality sign $$ \begin{aligned} &\text{Shaded region is above the solid boundary line.}\ &\text{Test point in shaded region: } (0, 0)\ &0 \ge 0^2 - 3 \implies 0 \ge -3 \quad (\text{True})\ &\text{Inequality: } y \ge x^2 - 3 \end{aligned} $$

(A) $y \ge x^2 - 3$

(B) $y \le x^2 + 3$

(C) $y \le x^2 - 3$

(D) $y \ge x^2 + 3$

{ "elements": [ { "type": "inequality", "params": [ { "js": "x*x - 3", "latex": "x^2 - 3" }, [0, 0] ], "properties": { "inverse": true, "strict": false, "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "point", "params": [[0, -3]], "properties": { "name": "Vertex (0, -3)", "size": 4, "color": "#583C87", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities in Two Variables" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the boundary parabola equation

Determine the inequality sign

Answer: