Question

plot each point, one at a time. using the graphing tool, start by plotting (0,0). use the graphing tool to plot (3, 0.5). use the graphing tool to plot (11, 1.0). use the graphing tool to plot (19, 1.3). use the graphing tool to plot (24, 1.4). is the data best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function? a. exponential function b. linear function c. logarithmic function d. quadratic function

Explanation:

A linear function has a constant rate of change (slope). An exponential function has a constant ratio between consecutive \(y\) - values. A logarithmic function has a slow - increasing or decreasing trend. A quadratic function has a parabolic shape.
Looking at the plotted points \((0,0)\), \((3,0.5)\), \((11,1.0)\), \((19,1.3)\), \((24,1.4)\), we can calculate the differences in \(x\) and \(y\) values.
The differences in \(x\) values: \(\Delta x_1=3 - 0=3\), \(\Delta x_2 = 11 - 3 = 8\), \(\Delta x_3=19 - 11 = 8\), \(\Delta x_4=24 - 19 = 5\)
The differences in \(y\) values: \(\Delta y_1=0.5 - 0=0.5\), \(\Delta y_2=1.0 - 0.5 = 0.5\), \(\Delta y_3=1.3 - 1.0=0.3\), \(\Delta y_4=1.4 - 1.3 = 0.1\)
The ratios of \(y\) - values: \(\frac{y_2}{y_1}=\frac{0.5}{0}=undefined\) (not exponential).
For a linear function \(y=mx + b\), if we assume \(b = 0\) (since when \(x = 0,y = 0\)), \(m=\frac{y}{x}\). For \((x = 3,y = 0.5)\), \(m=\frac{0.5}{3}\approx0.17\); for \((x = 11,y = 1.0)\), \(m=\frac{1.0}{11}\approx0.09\); for \((x = 19,y = 1.3)\), \(m=\frac{1.3}{19}\approx0.07\); for \((x = 24,y = 1.4)\), \(m=\frac{1.4}{24}\approx0.06\). The slope is not constant.
A logarithmic function \(y=a\ln(x)+b\) (or \(y = a\log(x)+b\)) has a non - constant rate of change that slows down as \(x\) increases. The general shape of the points (the rate of increase of \(y\) with respect to \(x\) is slowing down) is characteristic of a logarithmic function.

Answer:

C. logarithmic function