select the graph that correctly represents $f(x) = \frac{1}{2}(x - 3)^2 - 6$.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Parabola Vertex Form", "Parabola Features", "Quadratic y-intercept" ], "new_concepts": [], "current_concepts": [ "Parabola Vertex Form", "Parabola Features", "Quadratic y-intercept" ] } ## Identify the vertex of the parabola Using the *Parabola Vertex Form* knowledge point $$ \begin{aligned} &f(x) = a(x-h)^2 + k\ &a = \frac{1}{2},\quad h = 3,\quad k = -6\ &\text{Vertex } (h, k) = (3, -6) \end{aligned} $$ ## Determine key features of the graph Using the *Parabola Features* knowledge point $$ \begin{aligned} &a = \frac{1}{2} > 0 \implies \text{Opens upward}\ &\text{Axis of symmetry: } x = 3 \end{aligned} $$ ## Calculate the y-intercept Using the *Quadratic y-intercept* knowledge point $$ \begin{aligned} &f(0) = \frac{1}{2}(0-3)^2 - 6\ &f(0) = \frac{1}{2}(9) - 6 = 4.5 - 6 = -1.5\ &\text{y-intercept: } (0, -1.5) \end{aligned} $$ ## Find additional points on the parabola We calculate points on either side of the vertex to verify the graph: $$ \begin{aligned} &f(1) = \frac{1}{2}(1-3)^2 - 6 = \frac{1}{2}(4) - 6 = -4 \implies (1, -4)\ &f(5) = \frac{1}{2}(5-3)^2 - 6 = \frac{1}{2}(4) - 6 = -4 \implies (5, -4)\ &f(-1) = \frac{1}{2}(-1-3)^2 - 6 = \frac{1}{2}(16) - 6 = 2 \implies (-1, 2)\ &f(7) = \frac{1}{2}(7-3)^2 - 6 = \frac{1}{2}(16) - 6 = 2 \implies (7, 2) \end{aligned} $$ ## Match features with the given options - **Graph A**: The vertex is located at $(3, -6)$. The y-intercept is at $(0, -1.5)$. It passes through $(1, -4)$, $(5, -4)$, $(-1, 2)$, and $(7, 2)$. This perfectly matches our calculated features. - **Graph B**: The vertex is located at $(3, -8)$, which is incorrect. - **Graph C**: The vertex is located at $(-3, -6)$, which is incorrect. - **Graph D**: The vertex is cut off but clearly not at $(3, -6)$.

(A) Graph with vertex at (3, -6) and y-intercept at (0, -1.5)

(B) Graph with vertex at (3, -8)

(C) Graph with vertex at (-3, -6)

(D) Graph with vertex at (0, -6)

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "0.5 * Math.pow(x - 3, 2) - 6", "latex": "\frac{1}{2}(x-3)^2 - 6" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f(x)" } }, { "type": "point", "params": [ [3, -6] ], "properties": { "name": "Vertex (3, -6)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [0, -1.5] ], "properties": { "name": "y-int (0, -1.5)", "size": 4, "color": "#5583F2", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Parabola Vertex Form" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the vertex of the parabola

Determine key features of the graph

Calculate the y-intercept

Find additional points on the parabola

Match features with the given options

Answer: