Question

A table with certain points is shown.


x 2 −2 0
y 5 1 3


Part A: Choose two points from the table and calculate the slope between them. Show all necessary work. (4 points)

Part B: Choose two different points from the table and calculate the slope between them. Show all necessary work. (4 points)

Part C: What do the slopes from parts A and B tell you about the relationship between the points? Explain. (4 points)

Answer 1

Let's go through each part step by step.

### Part A:

**Choose two points from the table:**
Let's choose the points (2, 5) and (0, 3).

**Formula for the slope (m) between two points (x₁, y₁) and (x₂, y₂):**

\[
m = \frac{y₂ - y₁}{x₂ - x₁}
\]

**Assign values:**
- Point 1 (x₁, y₁) = (2, 5)
- Point 2 (x₂, y₂) = (0, 3)

**Calculate the slope:**
\[
m = \frac{3 - 5}{0 - 2} = \frac{-2}{-2} = 1
\]

So, the slope between the points (2, 5) and (0, 3) is **1**.

### Part B:

**Choose two different points from the table:**
Let's choose the points (−2, 1) and (0, 3).

**Assign values:**
- Point 1 (x₁, y₁) = (−2, 1)
- Point 2 (x₂, y₂) = (0, 3)

**Calculate the slope:**
\[
m = \frac{3 - 1}{0 - (-2)} = \frac{2}{2} = 1
\]

So, the slope between the points (−2, 1) and (0, 3) is **1**.

### Part C:

**Interpretation of the slopes from Parts A and B:**

Both slopes calculated in Parts A and B are equal to **1**. This indicates that the relationship between the points is linear with a consistent positive slope.

A slope of 1 means that for every unit increase in the x-direction, there is a corresponding increase of 1 unit in the y-direction. This consistency suggests that all points lie along a line with a slope of 1. In other words, as one variable increases, the other variable increases at the same rate, representing a direct proportional relationship between the x and y values.

### Summary:
- Part A slope: 1 (between points (2, 5) and (0, 3)).
- Part B slope: 1 (between points (−2, 1) and (0, 3)).
- Conclusion: The points demonstrate a linear relationship with a slope of 1, meaning a direct proportional increase in y for any increase in x.