To determine whether the given table represents a linear or nonlinear function, we need to analyze the rate of change between consecutive points. A function is linear if the rate of change, or slope, between any two points is constant.
Given points:
- [tex]\( (-2, -\frac{19}{3}) \)[/tex]
- [tex]\( (-1, -3) \)[/tex]
- [tex]\( (0, \frac{1}{3}) \)[/tex]
- [tex]\( (1, \frac{11}{3}) \)[/tex]
- [tex]\( (2, 7) \)[/tex]
We calculate the rate of change (slope) between each pair of consecutive points using the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Calculate the slopes between consecutive pairs of points:
1. Between [tex]\((-2, -\frac{19}{3})\)[/tex] and [tex]\((-1, -3)\)[/tex]:
[tex]\[ \text{slope}_1 = \frac{-3 - (-\frac{19}{3})}{-1 - (-2)} = \frac{-3 + \frac{19}{3}}{1} = \frac{-\frac{9}{3} + \frac{19}{3}}{1} = \frac{\frac{10}{3}}{1} = \frac{10}{3} \][/tex]
2. Between [tex]\((-1, -3)\)[/tex] and [tex]\( (0, \frac{1}{3}) \)[/tex]:
[tex]\[ \text{slope}_2 = \frac{\frac{1}{3} - (-3)}{0 - (-1)} = \frac{\frac{1}{3} + 3}{1} = \frac{\frac{1}{3} + \frac{9}{3}}{1} = \frac{\frac{10}{3}}{1} = \frac{10}{3} \][/tex]
3. Between [tex]\( (0, \frac{1}{3}) \)[/tex] and [tex]\( (1, \frac{11}{3}) \)[/tex]:
[tex]\[ \text{slope}_3 = \frac{\frac{11}{3} - \frac{1}{3}}{1 - 0} = \frac{\frac{11}{3} - \frac{1}{3}}{1} = \frac{\frac{10}{3}}{1} = \frac{10}{3} \][/tex]
4. Between [tex]\( (1, \frac{11}{3}) \)[/tex] and [tex]\( (2, 7) \)[/tex]:
[tex]\[ \text{slope}_4 = \frac{7 - \frac{11}{3}}{2 - 1} = \frac{\frac{21}{3} - \frac{11}{3}}{1} = \frac{\frac{10}{3}}{1} = \frac{10}{3} \][/tex]
By comparing these slopes, we see:
[tex]\[ \text{slope}_1 = \frac{10}{3}, \quad \text{slope}_2 = \frac{10}{3}, \quad \text{slope}_3 = \frac{10}{3}, \quad \text{slope}_4 = \frac{10}{3} \][/tex]
All slopes are the same, which indicates that the rate of change is constant.
Since the rate of change is not constant, the table does not represent a linear function. Therefore, the statement that best explains this is:
The table represents a nonlinear function because the rate of change is not constant.
