sketch the graph of the function.

(g(x) = (x - 2)^2 - 6)

use the graphing tool to graph the function.

Question

sketch the graph of the function.

(g(x) = (x - 2)^2 - 6)

use the graphing tool to graph the function.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Graphing Quadratic Functions", "Finding Vertex of Quadratic" ], "new_concepts": [], "current_concepts": [ "Graphing Quadratic Functions", "Finding Vertex of Quadratic", "Function Transformations" ] } ## Identify the vertex and key points of the parabola $$ \begin{aligned} &g(x) = (x-2)^2 - 6 \ &\text{Vertex: } (h, k) = (2, -6) \ &g(0) = (0-2)^2 - 6 = -2 \implies (0, -2) \ &g(1) = (1-2)^2 - 6 = -5 \implies (1, -5) \ &g(2) = -6 \implies (2, -6) \ &g(3) = (3-2)^2 - 6 = -5 \implies (3, -5) \ &g(4) = (4-2)^2 - 6 = -2 \implies (4, -2) \end{aligned} $$ ## Determine the x-intercepts $$ \begin{aligned} &(x-2)^2 - 6 = 0 \ &(x-2)^2 = 6 \ &x = 2 \pm \sqrt{6} \approx 2 \pm 2.45 \ &x_1 \approx 4.45, \quad x_2 \approx -0.45 \end{aligned} $$ To graph the function $g(x) = (x-2)^2 - 6$, plot the parabola with the following key features: - **Vertex**: $(2, -6)$ - **y-intercept**: $(0, -2)$ - **Other key points**: $(1, -5)$, $(3, -5)$, and $(4, -2)$ { "elements": [ { "type": "point", "params": [[2, -6]], "properties": { "name": "Vertex (2, -6)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -2]], "properties": { "name": "(0, -2)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[4, -2]], "properties": { "name": "(4, -2)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "Math.pow(x-2, 2) - 6", "latex": "(x-2)^2 - 6"}, -10, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Quadratic Functions" ] }

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Question

Explanation:

Identify the vertex and key points of the parabola

Determine the x-intercepts

Answer: