describe how the graph of the parent function $y = \sqrt{x}$ is transformed when graphing $y = -3\sqrt{x-6}$.
the graph is translated 6 units right.
the graph is reflected over the x-axis.
the graph has a vertical stretch by a factor of 3.

the function $y = -3\sqrt{x-6}$ is represented by graph
a.
b.
c.
d.

Question

describe how the graph of the parent function $y = \sqrt{x}$ is transformed when graphing $y = -3\sqrt{x-6}$.
the graph is translated 6 units right.
the graph is reflected over the x-axis.
the graph has a vertical stretch by a factor of 3.

the function $y = -3\sqrt{x-6}$ is represented by graph
a.
b.
c.
d.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Square Root Function Graphing", "Function Reflections", "Vertical Dilation", "Horizontal Translation" ], "new_concepts": [], "current_concepts": [ "Square Root Function Graphing", "Function Reflections", "Vertical Dilation", "Horizontal Translation" ] } ## Identify the transformations of the function Using the *Horizontal Translation*, *Function Reflections*, and *Vertical Dilation* knowledge points $$ \begin{aligned} &\text{Parent function: } y = \sqrt{x}\ &\text{Horizontal translation: } x \to x - 6 \implies \text{translated 6 units right}\ &\text{Vertical reflection: } \text{multiplied by } -1 \implies \text{reflected over the } x\text{-axis}\ &\text{Vertical dilation: } \text{multiplied by } 3 \implies \text{vertical stretch by a factor of 3} \end{aligned} $$ ## Determine the starting point and key features of the graph Using the *Square Root Function Graphing* knowledge point $$ \begin{aligned} &\text{Domain restriction: } x - 6 \ge 0 \implies x \ge 6\ &\text{Starting point: } (6, 0)\ &\text{Direction: } \text{Since } y = -3\sqrt{x-6} \le 0 \text{ for all } x \ge 6\text{, the graph curves downwards into Quadrant IV.} \end{aligned} $$ ## Evaluate a test point to distinguish between the graphs Using the *Square Root Function Graphing* knowledge point $$ \begin{aligned} &\text{Let } x = 10:\ &y = -3\sqrt{10-6} = -3\sqrt{4} = -3(2) = -6\ &\text{The correct graph must pass through } (10, -6). \end{aligned} $$ ## Match the test point with the given graph options Using the *Square Root Function Graphing* knowledge point $$ \begin{aligned} &\text{Graph A (blue curve): passes through } (10, -6).\ &\text{Graph B (orange curve): passes through } (10, -3).\ &\text{Graph C (green curve): lies above the } x\text{-axis (not reflected).}\ &\text{Graph D (purple curve): lies above the } x\text{-axis (not reflected).} \end{aligned} $$

(A) Graph A

(B) Graph B

(C) Graph C

(D) Graph D

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Square Root Function Graphing" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the transformations of the function

Determine the starting point and key features of the graph

Evaluate a test point to distinguish between the graphs

Match the test point with the given graph options

Answer: