Question

Plot the points (−2,2) and (4,4) in a rectangular coordinate​ system,​ (b) draw a line through the​ points, and ​(c) find and interpret the slope of the line.

Answer 1

To solve the problem, let's follow the steps laid out:

### (a) Plot the points (−2, 2) and (4, 4)

1. **Point (−2, 2)**:
- Move 2 units left on the x-axis (since it's -2).
- Move 2 units up on the y-axis (since it's +2).
- Mark this point.

2. **Point (4, 4)**:
- Move 4 units right on the x-axis.
- Move 4 units up on the y-axis.
- Mark this point.

### (b) Draw a line through the points

- Once both points are plotted, draw a straight line connecting them. This is the line that passes through both points.

### (c) Find and interpret the slope of the line

To find the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\), you can use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((−2, 2)\) and \((4, 4)\):
- \( (x_1, y_1) = (-2, 2) \)
- \( (x_2, y_2) = (4, 4) \)

Substituting into the formula:
\[
m = \frac{4 - 2}{4 - (-2)} = \frac{2}{4 + 2} = \frac{2}{6} = \frac{1}{3}
\]

### Interpretation of the slope:
The slope \( \frac{1}{3} \) means that for every 1 unit increase in \( y \), there is a corresponding increase of 3 units in \( x \). In other words, the line rises 1 unit vertically for every 3 units it moves horizontally. This indicates a positive relationship between the \( x \) and \( y \) values on this line.

In summary:
- Points plotted: \((-2, 2)\) and \((4, 4)\).
- Line drawn through these points.
- Slope calculated as \( \frac{1}{3} \), indicating a gentle upward trend as \( x \) increases.