Part A:
Given the two points A(3, -2) and B(7, 4),
Find the slope of the line that passes through these two points.
Explain the formula for calculating slope and show each step in detail.
Describe in words what the slope value represents in this case (e.g., direction, steepness, positive/negative, etc.).
Part B:
Suppose point B is moved to a new position, B'(7, -2).
What is the new slope of the line between A(3, -2) and B'(7, -2)?
How does this new slope compare to the one in Part A?
What kind of line is formed in this case (e.g., horizontal, vertical, diagonal)? Please explain why.
Full Answer Section
- Step 3: Calculate the change in x (x2−x1). Change in x = 7−3=4
- Step 4: Apply the slope formula.m=46
- Step 5: Simplify the fraction.m=23
Therefore, the slope of the line that passes through points A(3, -2) and B(7, 4) is 23.
3. Describe in words what the slope value represents:
In this case, the slope value of
23 represents the following:
- Direction: Since the slope is positive, the line is increasing from left to right. This means that as you move along the line from left to right, the y-values are increasing.
- Steepness: The magnitude of the slope (3/2 or 1.5) indicates the steepness of the line. For every 2 units moved horizontally to the right, the line rises 3 units vertically upwards. A larger absolute value of the slope indicates a steeper line.
- Positive/Negative: As mentioned, the slope is positive, confirming the upward trend of the line.
Part B: Analyzing a New Slope
Given points: A = (3, -2) B' = (7, -2)
1. What is the new slope of the line between A(3, -2) and B'(7, -2)?
Using the slope formula with A
(x1,y1)=(3,−2) and B'
(x2,y2)=(7,−2):
- Step 1: Identify the coordinates.x1=3y1=−2x2=7y2=−2
- Step 2: Calculate the change in y (y2−y1). Change in y = −2−(−2)=−2+2=0
- Step 3: Calculate the change in x (x2−x1). Change in x = 7−3=4
- Step 4: Apply the slope formula.m=40
- Step 5: Simplify the fraction.m=0
The new slope of the line between A(3, -2) and B'(7, -2) is 0.
2. How does this new slope compare to the one in Part A?
The new slope (0) is significantly different from the slope in Part A (
23 or 1.5).
- The slope in Part A was positive, indicating an upward-sloping line.
- The new slope is zero, indicating a horizontal line.
3. What kind of line is formed in this case? Please explain why.
In this case, a
horizontal line is formed.
Explanation:
A horizontal line has a slope of zero because there is no vertical change (rise) between any two points on the line. As shown in the calculation, the y-coordinates of points A and B' are the same (-2). This means that
y2−y1=0, resulting in a numerator of zero in the slope formula. Since the change in x is non-zero (4), the overall slope becomes 0. Graphically, this means the line runs perfectly flat across the coordinate plane.