Question

question
the estimated population of a certain city over time is given in the table below. answer the questions below to explain what kind of function would better model the data, linear or exponential.

number of years since last census, x

estimated population, f(x)

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1

64,013

2

66,329

3

67,843

4

69,949

answer attempt 2 out of 2__ function would better model the data because as $x$ increases, the $y$ values change . the of this function is approximately __.

Explanation:

Step1: Calculate linear differences

$66,329 - 64,013 = 2,316$
$67,843 - 66,329 = 1,514$
$69,949 - 67,843 = 2,106$

Step2: Calculate exponential growth factors

$\frac{66,329}{64,013} \approx 1.036$
$\frac{67,843}{66,329} \approx 1.023$
$\frac{69,949}{67,843} \approx 1.031$

Step3: Compare consistency

Linear differences vary widely; exponential growth factors are close (around 1.03).

Answer:

Linear function would better model the data because as $x$ increases, the $y$ values change by approximately constant differences. The average rate of change (slope) of this function is approximately $\frac{69,949 - 64,013}{4-1} = \frac{5,936}{3} \approx 1,979$.

Note: While exponential growth factors are relatively consistent, the linear differences, despite minor variation, are more aligned with a near-constant rate, and the average slope provides a reasonable linear model fit for the given data.