To determine the best description for the information in the table, we will analyze the possible linear relationships between consecutive points. Let's examine each pair of consecutive points and determine the linear function (if it exists).
### Analyzing Consecutive Points
1. Points (-2, 3) and (-1, -1)
- [tex]\((x_1, y_1) = (-2, 3)\)[/tex]
- [tex]\((x_2, y_2) = (-1, -1)\)[/tex]
- Slope:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{-1 - (-2)} = \frac{-4}{1} = -4
\][/tex]
- Intercept:
[tex]\[
b = y_1 - mx_1 = 3 - (-4)(-2) = 3 - 8 = -5
\][/tex]
- Linear function:
[tex]\[
y = -4x - 5
\][/tex]
2. Points (-1, -1) and (0, -5)
- [tex]\((x_1, y_1) = (-1, -1)\)[/tex]
- [tex]\((x_2, y_2) = (0, -5)\)[/tex]
- Slope:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - (-1)}{0 - (-1)} = \frac{-4}{1} = -4
\][/tex]
- Intercept:
[tex]\[
b = y_1 - mx_1 = -1 - (-4)(-1) = -1 - 4 = -5
\][/tex]
- Linear function:
[tex]\[
y = -4x - 5
\][/tex]
3. Points (0, -5) and (-1, -9)
- [tex]\((x_1, y_1) = (0, -5)\)[/tex]
- [tex]\((x_2, y_2) = (-1, -9)\)[/tex]
- Slope:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-9 - (-5)}{-1 - 0} = \frac{-4}{-1} = 4
\][/tex]
- Intercept:
[tex]\[
b = y_1 - mx_1 = -5 - 4(0) = -5
\][/tex]
- Linear function:
[tex]\[
y = 4x - 5
\][/tex]
4. Points (-1, -9) and (-2, -13)
- [tex]\((x_1, y_1) = (-1, -9)\)[/tex]
- [tex]\((x_2, y_2) = (-2, -13)\)[/tex]
- Slope:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-13 - (-9)}{-2 - (-1)} = \frac{-4}{-1} = 4
\][/tex]
- Intercept:
[tex]\[
b = y_1 - mx_1 = -9 - 4(-1) = -9 + 4 = -5
\][/tex]
- Linear function:
[tex]\[
y = 4x - 5
\][/tex]
### Summary
The descriptions of the linear functions between consecutive points are:
- Points (-2, 3) and (-1, -1): [tex]\( y = -4x - 5 \)[/tex]
- Points (-1, -1) and (0, -5): [tex]\( y = -4x - 5 \)[/tex]
- Points (0, -5) and (-1, -9): [tex]\( y = 4x - 5 \)[/tex]
- Points (-1, -9) and (-2, -13): [tex]\( y = 4x - 5 \)[/tex]
Therefore, the best description for the information in the table is that the data points exhibit linear behavior with different slopes for intervals between points: [tex]\(-4\)[/tex] initially, then switching to [tex]\(4\)[/tex]. The functions identified describe these linear relationships accurately.
