choose the graph of $y x^2 - 9$

Question

choose the graph of $y > x^2 - 9$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Quadratic Functions", "Quadratic Graphs", "Zeros of Quadratic Functions" ], "new_concepts": [ "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ], "current_concepts": [ "Quadratic Functions", "Quadratic Graphs", "Zeros of Quadratic Functions", "Quadratic Inequalities in Two Variables", "Graphing Inequalities" ] } ## Identify the boundary parabola and its properties $$ \begin{aligned} &\text{Boundary Equation: } y = x^2 - 9\ &\text{Vertex: } (0, -9)\ &\text{Zeros: } x^2 - 9 = 0 \implies x = \pm 3\ &\text{Direction: } \text{Opens upward (since the coefficient of } x^2 \text{ is positive)} \end{aligned} $$ ## Determine the boundary line style and shaded region $$ \begin{aligned} &\text{Inequality: } y > x^2 - 9\ &\text{Boundary Style: } \text{Dashed line (since the inequality is strict, } >)\ &\text{Shading Region: } \text{Above the parabola (since } y > \text{)} \end{aligned} $$ ## Match with the given options $$ \begin{aligned} &\text{Option 1: Parabola opens upward, vertex at } (0, -9)\text{, dashed boundary, shaded inside/above.}\ &\text{Option 2: Parabola opens downward.}\ &\text{Option 3: Parabola opens upward, vertex at } (0, -9)\text{, dashed boundary, shaded outside/below.}\ &\text{Option 4: Parabola opens downward.}\ &\implies \text{Correct choice is Option 1 (the first graph from the left).} \end{aligned} $$

(A) Dashed parabola opening upward with vertex at (0, -9), shaded inside/above

(B) Dashed parabola opening downward with vertex at (0, 9), shaded below

(C) Dashed parabola opening upward with vertex at (0, -9), shaded outside/below

(D) Solid parabola opening downward with vertex at (0, 9), shaded inside

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Question

Explanation:

Identify the boundary parabola and its properties

Determine the boundary line style and shaded region

Match with the given options

Answer: