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.
Part B: Choose two different points from the table and calculate the slope between them. Show all necessary work.
Part C: What do the slopes from parts A and B tell you about the relationship between the points? Explain.

Answers

Answered by GPT-4o mini
Let's start with the given points:

| x | y |
|-----|-----|
| 2 | 5 |
| -2 | 1 |
| 0 | 3 |

### Part A
Let's choose the points (2, 5) and (-2, 1) to calculate the slope between them.

The formula for calculating the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Assign the points:
- Point 1: \((x_1, y_1) = (2, 5)\)
- Point 2: \((x_2, y_2) = (-2, 1)\)

Now substitute into the formula:

\[
m = \frac{1 - 5}{-2 - 2} = \frac{-4}{-4} = 1
\]

### Part B
Now, let's select the points (0, 3) and (2, 5) to calculate the slope between them.

Using the same slope formula:

- Point 1: \((x_1, y_1) = (0, 3)\)
- Point 2: \((x_2, y_2) = (2, 5)\)

Now substitute into the formula:

\[
m = \frac{5 - 3}{2 - 0} = \frac{2}{2} = 1
\]

### Part C
The slopes calculated in parts A and B are both equal to 1. This indicates that there is a consistent linear relationship between the points chosen. Specifically, a slope of 1 means that for every unit increase in \(x\), there is a corresponding unit increase in \(y\).

Since both slopes are equal, it suggests that the points lie on a straight line with equal vertical and horizontal changes, indicating a positive correlation. In other words, as \(x\) increases, \(y\) also increases at a constant rate.
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