what is the range of $y = \sqrt{x + 7} + 5$?

$y \ge -5$

$y \ge 5$

$y \ge -7$

all real numbers

Question

what is the range of $y = \sqrt{x + 7} + 5$?

$y \ge -5$

$y \ge 5$

$y \ge -7$

all real numbers

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Square Root Transformations" ], "new_concepts": [ "Range of Square Root Functions" ], "current_concepts": [ "Square Root Transformations", "Range of Square Root Functions" ] } ## Analyze the parent function Using the *Square Root Transformations* knowledge point $$ \begin{aligned} &f(x) = \sqrt{x}\ &\text{Domain: } x \ge 0\ &\text{Range: } f(x) \ge 0 \end{aligned} $$ ## Apply vertical translation Using the *Square Root Transformations* knowledge point $$ \begin{aligned} &y = \sqrt{x+7} + 5\ &\sqrt{x+7} \ge 0 \quad \text{for all } x \ge -7\ &y \ge 0 + 5 \implies y \ge 5 \end{aligned} $$

$y \ge -5$

$y \ge 5$

$y \ge -7$

all real numbers

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.sqrt(x + 7) + 5", "latex": "\sqrt{x+7}+5" }, -7, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "y = \sqrt{x+7}+5", "withLabel": true } }, { "type": "point", "params": [ [-7, 5] ], "properties": { "name": "(-7, 5)", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "line", "params": [ [-10, 5], [10, 5] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2, "name": "y = 5", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Range of Square Root Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the parent function

Apply vertical translation

Answer: