which properties are present in a table that represents an exponential function in the form $y = b^x$ when $b 1$?
i. as the $x$-values increase, the $y$-values increase.
ii. the point $(1, 0)$ exists in the table.
iii. as the $x$-values increase, the $y$-values decrease.
iv. as the $x$-values decrease, the $y$-values decrease, approaching a singular value.

i and iv
i and ii
ii and iii
iii only

Question

which properties are present in a table that represents an exponential function in the form $y = b^x$ when $b > 1$?
i. as the $x$-values increase, the $y$-values increase.
ii. the point $(1, 0)$ exists in the table.
iii. as the $x$-values increase, the $y$-values decrease.
iv. as the $x$-values decrease, the $y$-values decrease, approaching a singular value.

i and iv
i and ii
ii and iii
iii only

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Exponential Growth", "Asymptotic Behavior" ], "current_concepts": [ "Exponential Growth", "Asymptotic Behavior", "Exponential Functions" ] } ## Analyze statements I and III $$ \begin{aligned} &y = b^x \quad (b > 1) \ &\frac{dy}{dx} = b^x \ln(b) > 0 \implies \text{strictly increasing} \ &\text{Statement I is true; Statement III is false.} \end{aligned} $$ ## Analyze statement II $$ \begin{aligned} &y(1) = b^1 = b eq 0 \quad (\text{since } b > 1) \ &\text{Statement II is false.} \end{aligned} $$ ## Analyze statement IV $$ \begin{aligned} &\lim_{x \to -\infty} b^x = 0 \ &\text{As } x \text{ decreases, } y \text{ decreases and approaches } 0. \ &\text{Statement IV is true.} \end{aligned} $$

I and IV

I and II

II and III

III only

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze statements I and III

Analyze statement II

Analyze statement IV

Answer: