To calculate the slope between two points, we use the formula:
\[ \text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \]
Part A:
Let's choose the points (0, 4) and (-2, 2).
- Here, \(x_1 = 0\), \(y_1 = 4\), \(x_2 = -2\), \(y_2 = 2\).
Now, we can plug these values into the slope formula:
\[ m = \frac{2 - 4}{-2 - 0} = \frac{-2}{-2} = 1 \]
Slope from Part A: The slope between the points (0, 4) and (-2, 2) is 1.
Part B:
Now we will choose a different pair of points, specifically (-2, 2) and (-4, 1).
- Here, \(x_1 = -2\), \(y_1 = 2\), \(x_2 = -4\), \(y_2 = 1\).
Now, we apply the slope formula again:
\[ m = \frac{1 - 2}{-4 - (-2)} = \frac{-1}{-2} = \frac{1}{2} \]
Slope from Part B: The slope between the points (-2, 2) and (-4, 1) is 1/2.
Part C:
The slopes calculated from parts A and B are 1 and 1/2, respectively.
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The slope of 1 from Part A indicates that the relationship between the points (0, 4) and (-2, 2) is a linear relationship where for every unit decrease in \(x\), \(y\) decreases by the same unit. This means that the line connecting these two points is at a 45-degree angle to the x-axis.
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The slope of 1/2 from Part B indicates that the relationship between the points (-2, 2) and (-4, 1) is such that for every unit decrease in \(x\), \(y\) decreases by half a unit. This means that the line connecting these two points is less steep than the line connecting the points in Part A.
The differing slopes suggest that while both sets of points are linear, they represent different rates of change in \(y\) as \(x\) changes. Specifically, the first pair of points (0, 4) and (-2, 2) descends faster than the second pair (-2, 2) and (-4, 1). This suggests that the relationship is not constant across the entire set of points; the rate of change in \(y\) with respect to changes in \(x\) varies as we move along the chosen points.