which is a feature of function $g$ if $g(x) = -f(x - 4)$?

$x$-intercept at $(5, 0)$
$y$-intercept at $(0, -4)$
vertical asymptote of $x = -4$
horizontal asymptote of $y = 4$

Question

which is a feature of function $g$ if $g(x) = -f(x - 4)$?

$x$-intercept at $(5, 0)$
$y$-intercept at $(0, -4)$
vertical asymptote of $x = -4$
horizontal asymptote of $y = 4$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Functions", "Function Domain", "Function Intercepts", "Asymptotic Behavior" ], "new_concepts": [ "Function Transformations", "Horizontal Translation", "Reflection Transformation" ], "current_concepts": [ "Logarithmic Functions", "Function Domain", "Function Intercepts", "Asymptotic Behavior", "Function Transformations", "Horizontal Translation", "Reflection Transformation" ] } ## Identify key features of the parent function $f(x)$ $$ \begin{aligned} &\text{Vertical asymptote: } x = 0\ &\text{x-intercept: } (1, 0)\ &\text{y-intercept: } \text{None (asymptote at } x = 0\text{)}\ &\text{Key point: } (3, 1) \end{aligned} $$ ## Apply transformations to find $g(x) = -f(x - 4)$ $$ \begin{aligned} &\text{Horizontal shift: } 4 \text{ units right } (x \to x + 4)\ &\text{Vertical reflection: } \text{reflect across the x-axis } (y \to -y)\ &\text{New vertical asymptote: } x = 0 + 4 \implies x = 4\ &\text{New x-intercept: } (1 + 4, -0) \implies (5, 0) \end{aligned} $$ ## Evaluate the given options $$ \begin{aligned} &\text{Option 1: x-intercept at } (5, 0) \quad \text{(True)}\ &\text{Option 2: y-intercept at } (0, -4) \quad \text{(False, domain is } x > 4\text{)}\ &\text{Option 3: vertical asymptote of } x = -4 \quad \text{(False, asymptote is } x = 4\text{)}\ &\text{Option 4: horizontal asymptote of } y = 4 \quad \text{(False, logarithmic functions have no horizontal asymptote)} \end{aligned} $$

x-intercept at (5, 0)

y-intercept at (0,-4)

vertical asymptote of $x = -4$

horizontal asymptote of $y = 4$

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

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Question

Explanation:

Identify key features of the parent function \(f(x)\)

Apply transformations to find \(g(x) = -f(x - 4)\)

Evaluate the given options

Answer: