graph the function using a combination of transformations applied to the graph of a basic function.

(g(x) = 5(x + 3)^2 - 1)

Question

graph the function using a combination of transformations applied to the graph of a basic function.

(g(x) = 5(x + 3)^2 - 1)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Graphing Parabolas", "Horizontal and Vertical Shifts", "Reflections and Stretching" ], "new_concepts": [], "current_concepts": [ "Graphing Parabolas", "Horizontal and Vertical Shifts", "Reflections and Stretching" ] } ## Identify the vertex and transformations $$ \begin{aligned} &g(x) = 5(x + 3)^2 - 1\ &\text{Vertex } (h, k) = (-3, -1)\ &\text{Horizontal shift: } 3 \text{ units left}\ &\text{Vertical shift: } 1 \text{ unit down}\ &\text{Vertical stretch factor: } 5 \end{aligned} $$ ## Calculate key points on the parabola $$ \begin{aligned} &x = -3 \implies g(-3) = -1 \implies (-3, -1)\ &x = -2 \implies g(-2) = 5(-2+3)^2 - 1 = 4 \implies (-2, 4)\ &x = -4 \implies g(-4) = 5(-4+3)^2 - 1 = 4 \implies (-4, 4) \end{aligned} $$ ## Determine the graph features $$ \begin{aligned} &\text{Vertex: } (-3, -1)\ &\text{Points: } (-4, 4), (-2, 4) \end{aligned} $$ To graph the function $g(x) = 5(x + 3)^2 - 1$, plot the vertex at $(-3, -1)$ and at least one other point, such as $(-2, 4)$ or $(-4, 4)$, then draw the upward-opening parabola passing through these points. { "elements": [ { "type": "point", "params": [[-3, -1]], "properties": { "name": "Vertex (-3, -1)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-2, 4]], "properties": { "name": "(-2, 4)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-4, 4]], "properties": { "name": "(-4, 4)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "5*Math.pow(x + 3, 2) - 1", "latex": "5(x+3)^2-1"}, -5, -1], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Parabolas" ] }

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Question

Explanation:

Identify the vertex and transformations

Calculate key points on the parabola

Determine the graph features

Answer: