which statement is true?

- the graph of $y = \log_b(x-4)$ is the graph of $y = \log_b(x)$ translated 4 units down.
- the graph of $y = \log_b(x)-4$ is the graph of $y = \log_b(x)$ translated 4 units left.
- the graph of $y = \log_b(x)+4$ is the graph of $y = \log_b(x)$ translated 4 units up.
- the graph of $y = \log_b(x+4)$ is the graph of $y = \log_b(x)$ translated 4 units right.

Question

which statement is true?

- the graph of $y = \log_b(x-4)$ is the graph of $y = \log_b(x)$ translated 4 units down.
- the graph of $y = \log_b(x)-4$ is the graph of $y = \log_b(x)$ translated 4 units left.
- the graph of $y = \log_b(x)+4$ is the graph of $y = \log_b(x)$ translated 4 units up.
- the graph of $y = \log_b(x+4)$ is the graph of $y = \log_b(x)$ translated 4 units right.

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": [ "Logarithmic Function Transformations" ], "new_concepts": [], "current_concepts": [ "Logarithmic Function Transformations", "Horizontal Translation", "Vertical Translation" ] } ## Analyze horizontal translations Using the *Logarithmic Function Transformations* knowledge point $$ \begin{aligned} &y = \log_b(x - h) \implies \text{horizontal translation by } h \text{ units}\ &y = \log_b(x - 4) \implies \text{translated } 4 \text{ units right}\ &y = \log_b(x + 4) \implies \text{translated } 4 \text{ units left} \end{aligned} $$ ## Analyze vertical translations Using the *Logarithmic Function Transformations* knowledge point $$ \begin{aligned} &y = \log_b(x) + k \implies \text{vertical translation by } k \text{ units}\ &y = \log_b(x) - 4 \implies \text{translated } 4 \text{ units down}\ &y = \log_b(x) + 4 \implies \text{translated } 4 \text{ units up} \end{aligned} $$ ## Evaluate the given statements Using the *Logarithmic Function Transformations* knowledge point $$ \begin{aligned} &\text{Statement 1: } y = \log_b(x-4) \text{ is translated } 4 \text{ units down} \implies \text{False (it is right)}\ &\text{Statement 2: } y = \log_b(x)-4 \text{ is translated } 4 \text{ units left} \implies \text{False (it is down)}\ &\text{Statement 3: } y = \log_b(x)+4 \text{ is translated } 4 \text{ units up} \implies \text{True}\ &\text{Statement 4: } y = \log_b(x+4) \text{ is translated } 4 \text{ units right} \implies \text{False (it is left)} \end{aligned} $$

The graph of $y = \log_b(x-4)$ is the graph of $y = \log_b(x)$ translated 4 units down.

The graph of $y = \log_b(x)-4$ is the graph of $y = \log_b(x)$ translated 4 units left.

The graph of $y = \log_b(x)+4$ is the graph of $y = \log_b(x)$ translated 4 units up.

The graph of $y = \log_b(x+4)$ is the graph of $y = \log_b(x)$ translated 4 units right.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Logarithmic Function Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze horizontal translations

Analyze vertical translations

Evaluate the given statements

Answer: