Answer :
Sure, let's break down the solution step by step for Part D and Part E.
### Part D: Write the coordinates of two points on each line
Without the specific context or equations for Line A, Line B, and Line C, I will provide a generic framework. In general, to find two points on a line described by an equation [tex]\(y = mx + b\)[/tex] (where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept), you can select two values for [tex]\(x\)[/tex] and compute the corresponding [tex]\(y\)[/tex] values.
However, we need to imagine hypothetical data points for illustration purposes. Let's assume:
Line A:
Equation: [tex]\( y = 2x + 1 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0) + 1 = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2) + 1 = 5 \)[/tex]
Coordinates for Line A:
- Coordinate 1: [tex]\((0, 1)\)[/tex]
- Coordinate 2: [tex]\((2, 5)\)[/tex]
Line B:
Equation: [tex]\( y = 3x - 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 3(0) - 2 = -2 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = 3(2) - 2 = 4 \)[/tex]
Coordinates for Line B:
- Coordinate 1: [tex]\((0, -2)\)[/tex]
- Coordinate 2: [tex]\((2, 4)\)[/tex]
Line C:
Equation: [tex]\( y = -x + 4 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = - (0) + 4 = 4 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = - (2) + 4 = 2 \)[/tex]
Coordinates for Line C:
- Coordinate 1: [tex]\((0, 4)\)[/tex]
- Coordinate 2: [tex]\((2, 2)\)[/tex]
#### Summary Table:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & Line A & Line B & Line C \\ \hline Coordinate 1 & (0, 1) & (0, -2) & (0, 4) \\ \hline Coordinate 2 & (2, 5) & (2, 4) & (2, 2) \\ \hline \end{tabular} \][/tex]
What information does each point represent for the sponsor's pledge plan?
Each point represents the relationship between two quantities. For instance:
- In Line A, point (0, 1) signifies that when there are 0 units (e.g., distance, events, items, etc.), the output is 1. The point (2, 5) indicates that for 2 units, the output is 5.
- Similar interpretations can be made for the other lines.
### Part E: Does each relationship represent a proportional relationship?
A proportional relationship between two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is described by [tex]\(y = kx\)[/tex] where [tex]\(k\)[/tex] is a constant. This implies that the graph of the relationship is a straight line passing through the origin (0, 0).
- Line A: [tex]\( y = 2x + 1 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of 1, not 0.
- Line B: [tex]\( y = 3x - 2 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of -2, not 0.
- Line C: [tex]\( y = -x + 4 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of 4, not 0.
In conclusion, none of the relationships for Line A, Line B, and Line C represent a proportional relationship as they do not pass through the origin.
### Part D: Write the coordinates of two points on each line
Without the specific context or equations for Line A, Line B, and Line C, I will provide a generic framework. In general, to find two points on a line described by an equation [tex]\(y = mx + b\)[/tex] (where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept), you can select two values for [tex]\(x\)[/tex] and compute the corresponding [tex]\(y\)[/tex] values.
However, we need to imagine hypothetical data points for illustration purposes. Let's assume:
Line A:
Equation: [tex]\( y = 2x + 1 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0) + 1 = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2) + 1 = 5 \)[/tex]
Coordinates for Line A:
- Coordinate 1: [tex]\((0, 1)\)[/tex]
- Coordinate 2: [tex]\((2, 5)\)[/tex]
Line B:
Equation: [tex]\( y = 3x - 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 3(0) - 2 = -2 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = 3(2) - 2 = 4 \)[/tex]
Coordinates for Line B:
- Coordinate 1: [tex]\((0, -2)\)[/tex]
- Coordinate 2: [tex]\((2, 4)\)[/tex]
Line C:
Equation: [tex]\( y = -x + 4 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = - (0) + 4 = 4 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( y = - (2) + 4 = 2 \)[/tex]
Coordinates for Line C:
- Coordinate 1: [tex]\((0, 4)\)[/tex]
- Coordinate 2: [tex]\((2, 2)\)[/tex]
#### Summary Table:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & Line A & Line B & Line C \\ \hline Coordinate 1 & (0, 1) & (0, -2) & (0, 4) \\ \hline Coordinate 2 & (2, 5) & (2, 4) & (2, 2) \\ \hline \end{tabular} \][/tex]
What information does each point represent for the sponsor's pledge plan?
Each point represents the relationship between two quantities. For instance:
- In Line A, point (0, 1) signifies that when there are 0 units (e.g., distance, events, items, etc.), the output is 1. The point (2, 5) indicates that for 2 units, the output is 5.
- Similar interpretations can be made for the other lines.
### Part E: Does each relationship represent a proportional relationship?
A proportional relationship between two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is described by [tex]\(y = kx\)[/tex] where [tex]\(k\)[/tex] is a constant. This implies that the graph of the relationship is a straight line passing through the origin (0, 0).
- Line A: [tex]\( y = 2x + 1 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of 1, not 0.
- Line B: [tex]\( y = 3x - 2 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of -2, not 0.
- Line C: [tex]\( y = -x + 4 \)[/tex]
- This is not a proportional relationship because it has a y-intercept of 4, not 0.
In conclusion, none of the relationships for Line A, Line B, and Line C represent a proportional relationship as they do not pass through the origin.
