which of the following is the graph of this square root function?
\f(x) = \sqrt{x + 4} + 3\

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Graphing Radical Functions", "Radical Function Transformations", "Square Root Graphs" ], "new_concepts": [], "current_concepts": [ "Graphing Radical Functions", "Radical Function Transformations", "Square Root Graphs" ] } ## Identify the starting point of the function $$ \begin{aligned} &x + 4 \ge 0 \implies x \ge -4\ &f(-4) = \sqrt{-4 + 4} + 3 = 3\ &\text{Starting point: } (-4, 3) \end{aligned} $$ ## Evaluate key points on the graph $$ \begin{aligned} &f(-3) = \sqrt{-3 + 4} + 3 = 4 \implies (-3, 4)\ &f(0) = \sqrt{0 + 4} + 3 = 5 \implies (0, 5)\ &f(5) = \sqrt{5 + 4} + 3 = 6 \implies (5, 6) \end{aligned} $$ ## Match points with the given options $$ \begin{aligned} &\text{First graph starts at } (-3, 4) \text{ and passes through } (0, 5.5) \quad (\text{Incorrect})\ &\text{Second graph starts at } (-4, 3) \text{ and passes through } (0, 5) \text{ and } (5, 6) \quad (\text{Correct})\ &\text{Third graph starts at } (3, 3.5) \quad (\text{Incorrect}) \end{aligned} $$

(A) Graph starting at (-3, 4) with y-intercept near 5.5

(B) Graph starting at (-4, 3) with y-intercept at (0, 5)

(C) Graph starting at (3, 3.5) in the first quadrant

{ "elements": [ { "type": "point", "params": [[-4, 3]], "properties": { "name": "(-4, 3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 5]], "properties": { "name": "(0, 5)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[5, 6]], "properties": { "name": "(5, 6)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "Math.sqrt(x + 4) + 3", "latex": "\sqrt{x + 4} + 3" }, -4, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Radical Functions" ] }

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Question

Explanation:

Identify the starting point of the function

Evaluate key points on the graph

Match points with the given options

Answer: