a student is trying to determine if an exponential model is appropriate for the function $y = f(x)$ below.

$\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ y & 8.155 & 22.167 & 60.257 & 163.790 \\ \ln y & a & b & c & d \end{array}$

consider the positive constants $a$, $b$, $c$, and $d$ found by taking the natural log of the range values. what conclusions can be made about $y = f(x)$ based on this data?

- yes, an exponential model is appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase linearly.
- yes, an exponential model is appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase exponentially.
- no, an exponential model is not appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase linearly.
- no, an exponential model is not appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase exponentially.

Question

a student is trying to determine if an exponential model is appropriate for the function $y = f(x)$ below.

$\begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ y & 8.155 & 22.167 & 60.257 & 163.790 \\ \ln y & a & b & c & d \end{array}$

consider the positive constants $a$, $b$, $c$, and $d$ found by taking the natural log of the range values. what conclusions can be made about $y = f(x)$ based on this data?

- yes, an exponential model is appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase linearly.
- yes, an exponential model is appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase exponentially.
- no, an exponential model is not appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase linearly.
- no, an exponential model is not appropriate for $y = f(x)$ because as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase exponentially.

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": [ "Exponential Equations", "Logarithmic Models" ], "new_concepts": [ "Linearizing Exponential Data" ], "current_concepts": [ "Exponential Equations", "Logarithmic Models", "Linearizing Exponential Data" ] } ## Calculate natural logarithms of y Using the *Logarithmic Models* and *Exponential Equations* knowledge points $$ \begin{aligned} a &= \ln(8.155) \approx 2.0986 \ b &= \ln(22.167) \approx 3.0986 \ c &= \ln(60.257) \approx 4.0986 \ d &= \ln(163.790) \approx 5.0986 \end{aligned} $$ ## Analyze the rate of change Using the *Exponential Equations* knowledge point $$ \begin{aligned} b - a &= 3.0986 - 2.0986 = 1.0000 \ c - b &= 4.0986 - 3.0986 = 1.0000 \ d - c &= 5.0986 - 4.0986 = 1.0000 \end{aligned} $$ ## Determine model appropriateness Using the *Logarithmic Models* knowledge point $$ \begin{aligned} &\text{Since } \ln(y) \text{ increases by a constant amount of } 1.0 \text{ for each unit increase in } x,\ &\ln(y) \text{ is a linear function of } x.\ &\text{Therefore, an exponential model is appropriate for } y = f(x). \end{aligned} $$

(A) Yes, an exponential model is appropriate for $y = f(x)$ because the as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase linearly.

(B) Yes, an exponential model is appropriate for $y = f(x)$ because the as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.0986$, $b = 3.0986$, $c = 4.0986$, and $d = 5.0986$, increase exponentially.

(C) No, an exponential model is not appropriate for $y = f(x)$ because the as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase linearly.

(D) No, an exponential model is not appropriate for $y = f(x)$ because the as the $x$ values increase over equal-length intervals, the values of $\ln(y)$, $a = 2.100$, $b = 4.410$, $c = 9.261$, and $d = 19.448$, increase exponentially.

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

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QUESTION IMAGE

Question

Explanation:

Calculate natural logarithms of y

Analyze the rate of change

Determine model appropriateness

Answer: