draw a line for the axis of symmetry of function f. also mark the x-intercept(s), y-intercept, and vertex of the function.

$f(x) = -(x + 1)^2 + 4$

Question

draw a line for the axis of symmetry of function f. also mark the x-intercept(s), y-intercept, and vertex of the function.

$f(x) = -(x + 1)^2 + 4$

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" ] } ## Identify key features of the quadratic function $$ \begin{aligned} &f(x) = -(x + 1)^2 + 4 \ &\text{Vertex: } (h, k) = (-1, 4) \ &\text{Axis of Symmetry: } x = -1 \end{aligned} $$ ## Find the y-intercept $$ \begin{aligned} &f(0) = -(0 + 1)^2 + 4 = -1 + 4 = 3 \ &\text{y-intercept: } (0, 3) \end{aligned} $$ ## Find the x-intercepts $$ \begin{aligned} &-(x + 1)^2 + 4 = 0 \ &(x + 1)^2 = 4 \ &x + 1 = \pm 2 \ &x_1 = 1 \implies (1, 0) \ &x_2 = -3 \implies (-3, 0) \end{aligned} $$ * **Axis of Symmetry:** Vertical line $x = -1$ * **Vertex:** $(-1, 4)$ * **y-intercept:** $(0, 3)$ * **x-intercepts:** $(-3, 0)$ and $(1, 0)$ { "elements": [ { "type": "line", "params": [[-1, -10], [-1, 10]], "properties": { "name": "x = -1", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "point", "params": [[-1, 4]], "properties": { "name": "Vertex (-1, 4)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[0, 3]], "properties": { "name": "y-int (0, 3)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[-3, 0]], "properties": { "name": "x-int (-3, 0)", "size": 4, "color": "#55DDF2", "withLabel": true } }, { "type": "point", "params": [[1, 0]], "properties": { "name": "x-int (1, 0)", "size": 4, "color": "#55DDF2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "-Math.pow(x + 1, 2) + 4", "latex": "-(x+1)^2+4"}, -10, 10], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Parabola Key Features" ] }

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Explanation:

Identify key features of the quadratic function

Find the y-intercept

Find the x-intercepts

Answer: