in which quadrant(s) does the graph of (y = \sqrt{x} - 2) lie?

$o$ i and iv

$o$ iii and iv

Question

in which quadrant(s) does the graph of (y = \sqrt{x} - 2) lie?

$o$ i and iv

$o$ iii and iv

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Quadrants of Graphs", "Square Root Graphing" ], "current_concepts": [ "Square Root Function", "Domain of Functions", "Function Transformations", "Quadrants of Graphs", "Square Root Graphing" ] } ## Determine the domain and starting point $$ \begin{aligned} &\text{Domain: } x \ge 0\ &\text{Starting point (vertex): } (0, -2) \quad (\text{Quadrant IV}) \end{aligned} $$ ## Find the x-intercept $$ \begin{aligned} &\sqrt{x} - 2 = 0 \ &\sqrt{x} = 2 \ &x = 4 \quad \implies (4, 0) \end{aligned} $$ ## Identify the quadrants containing the graph $$ \begin{aligned} &\text{For } 0 \le x < 4: y < 0 \implies (x, y) \in \text{Quadrant IV} \ &\text{For } x > 4: y > 0 \implies (x, y) \in \text{Quadrant I} \end{aligned} $$

I and IV

III and IV

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QUESTION IMAGE

Question

Explanation:

Determine the domain and starting point

Find the x-intercept

Identify the quadrants containing the graph

Answer: