Question
Plot the points (−2,2) and (4,4) in a rectangular coordinate system and find and interpret the slope of the line.
Answers
GPT-4o mini
To plot the points \((-2, 2)\) and \((4, 4)\) in a rectangular coordinate system (also known as the Cartesian plane), you would follow these steps:
1. Start by locating the point \((-2, 2)\):
- Move 2 units to the left on the x-axis (negative direction) to get to \(-2\).
- Then move 2 units up on the y-axis to reach the point \(( -2, 2)\).
2. Next, locate the point \((4, 4)\):
- Move 4 units to the right on the x-axis to get to \(4\).
- Then move 4 units up on the y-axis to reach the point \(( 4, 4)\).
3. Mark both points on the graph.
Now to calculate the slope (m) of the line passing through these two points, you can use the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (-2, 2)\) and \((x_2, y_2) = (4, 4)\).
Substituting the coordinates into the slope formula:
\[
m = \frac{4 - 2}{4 - (-2)} = \frac{2}{4 + 2} = \frac{2}{6} = \frac{1}{3}
\]
So the slope of the line is \(\frac{1}{3}\).
### Interpretation of the Slope:
The slope measures the rate of change of \(y\) with respect to \(x\). Specifically, a slope of \(\frac{1}{3}\) means that for every 3 units you move horizontally (to the right) along the x-axis, the line rises by 1 unit vertically (upward). This indicates a positive correlation between the x and y values, showing that as \(x\) increases, \(y\) also increases.
1. Start by locating the point \((-2, 2)\):
- Move 2 units to the left on the x-axis (negative direction) to get to \(-2\).
- Then move 2 units up on the y-axis to reach the point \(( -2, 2)\).
2. Next, locate the point \((4, 4)\):
- Move 4 units to the right on the x-axis to get to \(4\).
- Then move 4 units up on the y-axis to reach the point \(( 4, 4)\).
3. Mark both points on the graph.
Now to calculate the slope (m) of the line passing through these two points, you can use the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (-2, 2)\) and \((x_2, y_2) = (4, 4)\).
Substituting the coordinates into the slope formula:
\[
m = \frac{4 - 2}{4 - (-2)} = \frac{2}{4 + 2} = \frac{2}{6} = \frac{1}{3}
\]
So the slope of the line is \(\frac{1}{3}\).
### Interpretation of the Slope:
The slope measures the rate of change of \(y\) with respect to \(x\). Specifically, a slope of \(\frac{1}{3}\) means that for every 3 units you move horizontally (to the right) along the x-axis, the line rises by 1 unit vertically (upward). This indicates a positive correlation between the x and y values, showing that as \(x\) increases, \(y\) also increases.