choose the graph of $y

Question

choose the graph of $y < -x^2 + 4x + 5$.

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", "Quadratic Inequalities" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Quadratic Graphs", "Quadratic Inequalities" ] } ## Analyze the boundary parabola $$ \begin{aligned} &y = -x^2 + 4x + 5\ &\text{Since } a = -1 < 0, \text{ the parabola opens downward.}\ &x_{\text{vertex}} = -\frac{b}{2a} = -\frac{4}{2(-1)} = 2\ &y_{\text{vertex}} = -(2)^2 + 4(2) + 5 = 9\ &\text{Vertex: } (2, 9)\ &y\text{-intercept: } (0, 5)\ &x\text{-intercepts: } -x^2 + 4x + 5 = 0 \implies -(x-5)(x+1) = 0 \implies x = 5, -1 \end{aligned} $$ ## Determine the inequality shading $$ \begin{aligned} &\text{Inequality: } y < -x^2 + 4x + 5\ &\text{The inequality uses } < \text{ (strict inequality), so the boundary is a dashed curve.}\ &\text{Test point } (0,0):\ &0 < -(0)^2 + 4(0) + 5 \implies 0 < 5 \quad (\text{True})\ &\text{The shaded region is below/inside the downward-opening parabola and contains } (0,0). \end{aligned} $$ ## Match with the given options $$ \begin{aligned} &\text{Option 1 (first graph): Downward-opening dashed parabola, vertex at } (2,9), \text{ shaded below/inside.}\ &\text{Option 2 (second graph): Upward-opening dashed parabola.}\ &\text{Option 3 (third graph): Downward-opening dashed parabola, shaded above/outside.}\ &\text{Option 4 (fourth graph): Upward-opening dashed parabola.}\ &\text{Therefore, the first option is correct.} \end{aligned} $$

(A) Downward-opening dashed parabola with vertex at $(2, 9)$, shaded below/inside containing the origin

(B) Upward-opening dashed parabola with vertex at $(2, -9)$, shaded above/inside

(C) Downward-opening dashed parabola with vertex at $(2, 9)$, shaded above/outside

(D) Upward-opening dashed parabola with vertex at $(2, -9)$, shaded below/outside

{ "elements": [ { "type": "inequality", "params": [ { "js": "-x*x + 4*x + 5", "latex": "y < -x^2 + 4x + 5" }, [2, 2] ], "properties": { "inverse": false, "strict": true, "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2 } }, { "type": "point", "params": [[2, 9]], "properties": { "name": "Vertex (2,9)", "size": 3, "color": "#583C87", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the boundary parabola

Determine the inequality shading

Match with the given options

Answer: