Question

for the data set shown by the table,
a. create a scatter plot for the data.
b. use the scatter plot to determine whether an
exponential function, logarithmic function, or a linear
function is the best choice for modeling the data.

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

eq9
eq15
eq25
eq30\)), 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.