which statements describe key features of function $g$ if $g(x) = f(x + 2)$?

$y$-intercept at $(0, 4)$
$y$-intercept at $(0, 1)$
horizontal asymptote of $y = 2$
horizontal asymptote of $y = 0$

Question

which statements describe key features of function $g$ if $g(x) = f(x + 2)$?

$y$-intercept at $(0, 4)$
$y$-intercept at $(0, 1)$
horizontal asymptote of $y = 2$
horizontal asymptote of $y = 0$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Function Transformations", "Exponential Function Graphs", "Asymptotes of Exponential Functions" ], "new_concepts": [], "current_concepts": [ "Function Transformations", "Exponential Function Graphs", "Asymptotes of Exponential Functions" ] } ## Identify key features of the parent function $f(x)$ $$ \begin{aligned} &\text{From the graph, } f(x) = 2^x\ &\text{y-intercept of } f(x): (0, 1)\ &\text{Point on } f(x): (2, 4)\ &\text{Horizontal asymptote of } f(x): y = 0 \end{aligned} $$ ## Apply the transformation to find $g(x)$ $$ \begin{aligned} &g(x) = f(x + 2) = 2^{x+2}\ &\text{This represents a horizontal shift left by 2 units.} \end{aligned} $$ ## Determine key features of $g(x)$ $$ \begin{aligned} &\text{y-intercept of } g(x): g(0) = f(2) = 4 \implies (0, 4)\ &\text{Horizontal asymptote of } g(x): y = 0 \quad (\text{unchanged by horizontal shift}) \end{aligned} $$

y-intercept at (0,4)

y-intercept at (0,1)

horizontal asymptote of y = 2

horizontal asymptote of y = 0

{ "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 the transformation to find \(g(x)\)

Determine key features of \(g(x)\)

Answer: