for the data set shown by the table,\na. create a scatter plot for the data.\nb. use the scatter plot to…

for the data set shown by the table,\na. create a scatter plot for the data.\nb. use the scatter plot to determine whether an\nexponential function, logarithmic function, or a linear\nfunction is the best choice for modeling the data.

for the data set shown by the table,\na. create a scatter plot for the data.\nb. use the scatter plot to determine whether an\nexponential function, logarithmic function, or a linear\nfunction is the best choice for modeling the data.

Answer

Explanation:

Step1: Create scatter plot

  • Let (x) represent the year. For simplicity, we can use (x = 2013) as (x = 0), (x=2014) as (x = 1), (x = 2015) as (x=2), (x = 2016) as (x = 3), (x=2017) as (x = 4), (x = 2018) as (x=5). Let (y) represent the number of books.
  • Plot the points ((0,26)), ((1,31)), ((2,40)), ((3,55)), ((4,80)), ((5,110)) on a coordinate - plane.

Step2: Analyze the shape of the scatter - plot

  • Calculate the differences between consecutive (y) - values:

    • (31−26 = 5)
    • (40−31=9)
    • (55 - 40=15)
    • (80−55 = 25)
    • (110−80=30)

  • Calculate the ratios of consecutive (y) - values:

    • (\frac{31}{26}\approx1.19)
    • (\frac{40}{31}\approx1.29)
    • (\frac{55}{40}=1.375)
    • (\frac{80}{55}\approx1.45)
    • (\frac{110}{80}=1.375)

  • For a linear function, the differences between consecutive (y) - values (the first - differences) are constant. For an exponential function, the ratios of consecutive (y) - values (the growth factor) are approximately constant.

  • The first - differences are not constant ((5\neq9\neq15\neq25\neq30)), and the ratios are not exactly constant but show a trend of growth that is more in line with an exponential - like growth (as opposed to a logarithmic function, which has a slower growth rate as (x) increases).

Answer:

a. Plot the points ((2013,26)), ((2014,31)), ((2015,40)), ((2016,55)), ((2017,80)), ((2018,110)) on a scatter - plot (with (x) - axis as year and (y) - axis as number of books). b. An exponential function is the best choice for modeling the data.