Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine which of the given tables shows a linear function, we need to check if the points in each table follow a consistent rate of change or slope. A linear function can be represented by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept. In other words, if we calculate the slope between each pair of points in the table and the slope remains constant, then the table represents a linear function.
### Analyzing the tables:
#### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 8 \\ \hline -1 & 2 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 6 \\ \hline \end{array} \][/tex]
To determine if this table shows a linear function, we calculate the slope between consecutive points:
1. Between [tex]\((-4, 8)\)[/tex] and [tex]\((-1, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 8}{-1 - (-4)} = \frac{-6}{3} = -2 \][/tex]
2. Between [tex]\((-1, 2)\)[/tex] and [tex]\( (1, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0 \][/tex]
3. Between [tex]\( (1, 2)\)[/tex] and [tex]\( (2, 4)\)[/tex]:
[tex]\[ \text{slope} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2 \][/tex]
4. Between [tex]\( (2, 4)\)[/tex] and [tex]\( (3, 6)\)[/tex]:
[tex]\[ \text{slope} = \frac{6 - 4}{3 - 2} = \frac{2}{1} = 2 \][/tex]
The slopes are not consistent (we have [tex]\(-2\)[/tex], [tex]\(0\)[/tex], and [tex]\(2\)[/tex]), so Table 1 does not show a linear function.
#### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -6 & -16 \\ \hline -2 & -8 \\ \hline 0 & -4 \\ \hline 1 & -2 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
1. Between [tex]\((-6, -16)\)[/tex] and [tex]\((-2, -8)\)[/tex]:
[tex]\[ \text{slope} = \frac{-8 - (-16)}{-2 - (-6)} = \frac{8}{4} = 2 \][/tex]
2. Between [tex]\((-2, -8)\)[/tex] and [tex]\( (0, -4)\)[/tex]:
[tex]\[ \text{slope} = \frac{-4 - (-8)}{0 - (-2)} = \frac{4}{2} = 2 \][/tex]
3. Between [tex]\( (0, -4)\)[/tex] and [tex]\( (1, -2)\)[/tex]:
[tex]\[ \text{slope} = \frac{-2 - (-4)}{1 - 0} = \frac{2}{1} = 2 \][/tex]
4. Between [tex]\( (1, -2)\)[/tex] and [tex]\( (3, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - (-2)}{3 - 1} = \frac{4}{2} = 2 \][/tex]
The slopes are consistent ([tex]\(2\)[/tex] between all consecutive points), so Table 2 does show a linear function.
#### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & -2 \\ \hline \end{array} \][/tex]
This table only contains a single point, which trivially represents a linear function (with slope undefined due to lack of another point).
### Conclusion:
Among the provided tables, Table 2 shows a linear function because the slope between all consecutive points is consistent.
### Analyzing the tables:
#### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & 8 \\ \hline -1 & 2 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 6 \\ \hline \end{array} \][/tex]
To determine if this table shows a linear function, we calculate the slope between consecutive points:
1. Between [tex]\((-4, 8)\)[/tex] and [tex]\((-1, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 8}{-1 - (-4)} = \frac{-6}{3} = -2 \][/tex]
2. Between [tex]\((-1, 2)\)[/tex] and [tex]\( (1, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0 \][/tex]
3. Between [tex]\( (1, 2)\)[/tex] and [tex]\( (2, 4)\)[/tex]:
[tex]\[ \text{slope} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2 \][/tex]
4. Between [tex]\( (2, 4)\)[/tex] and [tex]\( (3, 6)\)[/tex]:
[tex]\[ \text{slope} = \frac{6 - 4}{3 - 2} = \frac{2}{1} = 2 \][/tex]
The slopes are not consistent (we have [tex]\(-2\)[/tex], [tex]\(0\)[/tex], and [tex]\(2\)[/tex]), so Table 1 does not show a linear function.
#### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -6 & -16 \\ \hline -2 & -8 \\ \hline 0 & -4 \\ \hline 1 & -2 \\ \hline 3 & 2 \\ \hline \end{array} \][/tex]
1. Between [tex]\((-6, -16)\)[/tex] and [tex]\((-2, -8)\)[/tex]:
[tex]\[ \text{slope} = \frac{-8 - (-16)}{-2 - (-6)} = \frac{8}{4} = 2 \][/tex]
2. Between [tex]\((-2, -8)\)[/tex] and [tex]\( (0, -4)\)[/tex]:
[tex]\[ \text{slope} = \frac{-4 - (-8)}{0 - (-2)} = \frac{4}{2} = 2 \][/tex]
3. Between [tex]\( (0, -4)\)[/tex] and [tex]\( (1, -2)\)[/tex]:
[tex]\[ \text{slope} = \frac{-2 - (-4)}{1 - 0} = \frac{2}{1} = 2 \][/tex]
4. Between [tex]\( (1, -2)\)[/tex] and [tex]\( (3, 2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - (-2)}{3 - 1} = \frac{4}{2} = 2 \][/tex]
The slopes are consistent ([tex]\(2\)[/tex] between all consecutive points), so Table 2 does show a linear function.
#### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & -2 \\ \hline \end{array} \][/tex]
This table only contains a single point, which trivially represents a linear function (with slope undefined due to lack of another point).
### Conclusion:
Among the provided tables, Table 2 shows a linear function because the slope between all consecutive points is consistent.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
