To determine whether the data points represent a linear function, let's follow these steps:
1. Calculate the differences in the x-values (Δx) and y-values (Δy):
- First, find the differences between consecutive x-values:
- [tex]\( x_2 - x_1 = 3 - (-1) = 4 \)[/tex]
- [tex]\( x_3 - x_2 = 7 - 3 = 4 \)[/tex]
- [tex]\( x_4 - x_3 = 11 - 7 = 4 \)[/tex]
So, the differences in x-values are: [tex]\( [4, 4, 4] \)[/tex]
- Next, find the differences between consecutive y-values:
- [tex]\( y_2 - y_1 = 1 - (-2) = 3 \)[/tex]
- [tex]\( y_3 - y_2 = 4 - 1 = 3 \)[/tex]
- [tex]\( y_4 - y_3 = 7 - 4 = 3 \)[/tex]
So, the differences in y-values are: [tex]\( [3, 3, 3] \)[/tex]
2. Calculate the ratio of the differences (i.e., the slope between each pair of points):
- The ratio for each interval is calculated by dividing the difference in y-values by the difference in x-values:
- [tex]\( \frac{3}{4} = 0.75 \)[/tex]
- [tex]\( \frac{3}{4} = 0.75 \)[/tex]
- [tex]\( \frac{3}{4} = 0.75 \)[/tex]
So, the ratios are: [tex]\( [0.75, 0.75, 0.75] \)[/tex]
3. Check for consistency in the ratios:
- If the ratios (slopes) are the same for all intervals, the data points lie on a straight line, and therefore, the function is linear.
- In this case, all the ratios are 0.75, indicating that the data points lie on a line with a consistent slope of 0.75.
As all the ratios are the same, the given data represents:
[tex]\[ \text{a linear function} \][/tex]
