graph the function $f(x) = -x^2 - 4x - 11$.

plot the vertex. then plot another point on the parabola. if you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

Question

graph the function $f(x) = -x^2 - 4x - 11$.

plot the vertex. then plot another point on the parabola. if you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Quadratic Vertex", "Finding Parabola Points" ], "new_concepts": [], "current_concepts": [ "Quadratic Vertex", "Finding Parabola Points", "Graphing Quadratic Functions" ] } ## Find the vertex of the parabola Using the *Quadratic Vertex* knowledge point $$ \begin{aligned} &a = -1,\quad b = -4,\quad c = -11\ &x_v = -\frac{b}{2a} = -\frac{-4}{2(-1)} = -2\ &y_v = f(-2) = -(-2)^2 - 4(-2) - 11 = -4 + 8 - 11 = -7\ &\text{Vertex: } (-2, -7) \end{aligned} $$ ## Find another point on the parabola Using the *Finding Parabola Points* knowledge point $$ \begin{aligned} &x = -1\ &y = f(-1) = -(-1)^2 - 4(-1) - 11 = -1 + 4 - 11 = -8\ &\text{Point: } (-1, -8) \end{aligned} $$ To graph the function $f(x) = -x^2 - 4x - 11$: 1. **Vertex**: $(-2, -7)$ 2. **Another point**: $(-1, -8)$ (or $(-3, -8)$) { "elements": [ { "type": "point", "params": [[-2, -7]], "properties": { "name": "Vertex (-2, -7)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-1, -8]], "properties": { "name": "(-1, -8)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-3, -8]], "properties": { "name": "(-3, -8)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "-x*x - 4*x - 11", "latex": "-x^2 - 4x - 11"}, -10, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Quadratic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the vertex of the parabola

Find another point on the parabola

Answer: