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
1
2
3
4
Estimated Population, f(x)
48,461
40,153
31,845
23,503
Answer
Attempt 1 out of 2

function would better model the data because as x increases, the y values change . The of this function is approximately .
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Answer 1

This is best modeled by a linear equation: y = -6319x + 56,800, where x is the years since the last census, and 56,800 was the population of the census. The population is declining by 6319 each year.

Explanation:

Population growth equations are most often modeled with an exponential function. This situation is not. It is a linear function.

We can see this by creating a graph of the points. See the attached graph. The first thing to note is that all the points form a straight line. We know from this that an equation of the form y=mx+b can be found that described this population drop with time.

m, the sloe, can be determined by finding the Rise/Rub of the line. The Rise is the change in y for a change in x.

Let's choose two points to calculate the slope:

(1, 48,461) and

(4, 23,503)

The Rise is 23,503 - 48,461 = -24,958

The Run is 4 - 1 = 3

The slope is (-24,958)/3 or -8319

We can write y = -8319 + b, where b is the y-intercept. In this case, b respresents the population at the time of the census. We can find b by entering any of the data points in the above equation and solve for b:

y = -8319 + b

40153 = -8319*(2) + b for (2,40153)

b = 56,800

The linear equation describing this population decline is

y = -6319x + 56,800,

where x is the years since the last census.

The population is declining by 8319 per year since the census. The year the census was taken the population was 56,800.