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
a. draw the scatter plot. choose the correct graph below
o a.
o b.

Explanation:

Step1: Analyze the data points

The \(x -\)values (years) are \(2013,2014,2015,2016,2017,2018\) and the \(y -\)values (number of books) are \(26,33,45,56,80,108\).

Step2: Check the pattern in the scatter - plot

In a scatter - plot, if the data points follow a straight - line pattern (or approximately straight - line), it is modeled by a linear function. If the rate of change of \(y\) with respect to \(x\) is non - constant and the \(y\) values grow at an increasing rate (the difference between consecutive \(y\) values is increasing: \(33 - 26=7\), \(45 - 33 = 12\), \(56-45 = 11\), \(80 - 56=24\), \(108 - 80 = 28\)), it is not a linear function. For an exponential function \(y = a\cdot b^{x}\) (\(a
eq0,b>0,b
eq1\)), the ratio of consecutive \(y\) values is approximately constant. \(\frac{33}{26}\approx1.27\), \(\frac{45}{33}\approx1.36\), \(\frac{56}{45}\approx1.24\), \(\frac{80}{56}\approx1.43\), \(\frac{108}{80}=1.35\). The ratios are not exactly constant but are in a similar range. For a logarithmic function \(y=a + b\ln(x)\), the growth rate slows down as \(x\) increases.

Since the number of books is increasing at an increasing rate (not a constant rate as in linear and not a decreasing rate as in logarithmic), an exponential function is the best choice.

For part (a), when we plot the points \((2013,26)\), \((2014,33)\), \((2015,45)\), \((2016,56)\), \((2017,80)\), \((2018,108)\) on a scatter - plot, the \(x\) (year) values are increasing and the \(y\) (number of books) values are also increasing.
Let's assume we re - scale the \(x\) - axis. Let \(x = 2013\) correspond to \(x = 1\), \(x = 2014\) correspond to \(x = 2\), etc.
The point \((1,26)\), \((2,33)\), \((3,45)\), \((4,56)\), \((5,80)\), \((6,108)\)
In option A, as \(x\) (after re - scaling) increases, \(y\) increases. In option B, as \(x\) increases, \(y\) decreases.

Answer:

a. A.
b. An exponential function.