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 3 Years Since Last Census , 58,860 55,917 53,121 50,314 Estimated Population, f(x) Answer Attempt out of 2 function would better model the data because as x increases, the y value change The of this A linear An exponential nately
Based on the given information, it seems that the estimated population decreases over time. Therefore, a linear function would better model the data. This is because a linear function represents a constant rate of change, which aligns with the decreasing population trend. On the other hand, an exponential function represents a constant rate of growth or decay, which does not accurately reflect the decreasing population over time in this case.
For value change do I put additivly or multiplicativy
For the value change in a linear function, you would use an additive approach. This means that the change in the dependent variable (population in this case) would be added or subtracted by a constant amount as the independent variable (time in this case) increases. In contrast, for an exponential function, the value change is multiplicative, meaning that the dependent variable would increase or decrease by a constant rate or factor as the independent variable increases.
Ok so common difference/slope or common ratio/multiplier/base
That's correct! In a linear function, you would have a common difference or slope, which represents the constant rate of change in the dependent variable (population) as the independent variable (time) increases. This can be measured by calculating the difference between consecutive population values.
In an exponential function, you would have a common ratio or multiplier or base, which represents the constant rate of growth or decay. This can be found by calculating the quotient between consecutive population values.
In the given data, since the estimated population decreases over time, a linear function with a negative common difference or slope would better model the data.
The function is approximately
To determine the specific function that would model the data, we would need more specific information or additional data points. However, based on the given information of the estimated population for different years, a linear function is most appropriate. A linear function can be represented in the form: f(x) = mx + b, where m is the slope or the constant rate of change and b is the y-intercept.
To find the equation of the linear function, we need to determine the slope and the y-intercept. Given the population values:
Number of 3 Years Since Last Census: 0, 3, 6, 9 (x-values)
Estimated Population: 58,860, 55,917, 53,121, 50,314 (y-values)
We can find the slope using the formula: m = (change in y) / (change in x). Taking any two points, let's say (0, 58,860) and (3, 55,917), we have:
m = (55,917 - 58,860) / (3 - 0) = -981
Next, we can find the y-intercept (b) by substituting one of the points into the linear equation. Let's use (0, 58,860):
f(x) = mx + b
58,860 = -981(0) + b
b = 58,860
Therefore, the linear function that models the data is:
f(x) = -981x + 58,860