Question

use the graphing tool at the right to create a scatter plot for the data in the table. use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function.

x	y
1	- 2
2	0
3	4
4	12

plot each point, one at a time. using the graphing tool, start by plotting (0, - 3).

Explanation:

Step1: Plot points

Plot the points \((0, - 3)\), \((1,-2)\), \((2,0)\), \((3,4)\), \((4,12)\) on the given graph.

Step2: Analyze shape

As \(x\) increases, the \(y - values\) increase at an increasing rate. A linear function has a constant rate of change, which is not the case here. A logarithmic function has a slow - growing rate. This data does not follow a logarithmic pattern. Since the rate of change of \(y\) with respect to \(x\) is not constant, it is not linear. The data appears to be growing more rapidly, and an exponential function \(y = a\cdot b^{x}\) or a quadratic function \(y=ax^{2}+bx + c\) could be considered. But if we look at the differences between consecutive \(y - values\): \(-2-(-3)=1\), \(0 - (-2)=2\), \(4 - 0 = 4\), \(12-4 = 8\). The differences are not constant (so not linear), and they are doubling each time, which is characteristic of an exponential function.

Answer:

The data are best modeled by an exponential function.