plot the axis of symmetry and the point where the maximum value occurs for this function:

(h(x) = -(x + 2)^2 + 8).

Question

plot the axis of symmetry and the point where the maximum value occurs for this function:

(h(x) = -(x + 2)^2 + 8).

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Quadratic Functions", "Axis of Symmetry", "Parabola Key Features" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Axis of Symmetry", "Parabola Key Features", "Vertex of a Parabola" ] } ## Identify vertex and maximum value $$ \begin{aligned} &h(x) = -(x + 2)^2 + 8 \ &\text{Vertex form: } y = a(x - h)^2 + k \ &(h, k) = (-2, 8) \ &a = -1 < 0 \implies \text{Maximum occurs at } (-2, 8) \end{aligned} $$ ## Determine axis of symmetry $$ \begin{aligned} &x = h \ &x = -2 \end{aligned} $$ - **Axis of Symmetry:** The vertical line $x = -2$ - **Point of Maximum Value (Vertex):** $(-2, 8)$ { "elements": [ { "type": "line", "params": [[-2, -10], [-2, 10]], "properties": { "name": "x = -2", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "point", "params": [[-2, 8]], "properties": { "name": "Maximum (-2, 8)", "size": 5, "color": "#F2557F", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "-Math.pow(x + 2, 2) + 8", "latex": "-(x+2)^2+8"}, -10, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 1.5 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Parabola Key Features" ] }

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Explanation:

Identify vertex and maximum value

Determine axis of symmetry

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