Problem 11 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-2,1)\) and \((2,2)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((-2,1)\) and \((2,2)\) is \(\frac{1}{4}\) and the line rises.

Step 1: Identify the coordinates of the points

The two points given are \((-2,1)\) and \((2,2)\). Let's denote these as point A \((-2,1)= (x_1, y_1)\) and point B \((2,2) = (x_2, y_2)\).

Step 2: Use the slope formula

The formula for the slope is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. Substitute the given co-ordinates of the two points into the slope formula to calculate the slope.

Step 3: Calculate the slope

Substituting the coordinates into the formula we get: \[m= \frac{2 - 1}{2 - (-2)} = \frac{1}{4}\].

Step 4: Determine the direction of the line

Since the slope is positive, we can conclude that the line rises.

Key Concepts

Slope Formula

Understanding the slope formula is critical in coordinate geometry as it reveals much about the direction and steepness of a line. In its simplest form, the slope formula is defined as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) represents the slope of the line and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of any two points on the line. Applied to a real-world example, let's consider two points, A \( (-2, 1) \) and B \( (2, 2) \). By plugging these points into the slope formula, we find that the slope \( m \) equals \( \frac{1}{4} \). This indicates a positive slope, signifying that as we move from left to right, the line rises consistently at a moderate incline. For students looking to master this concept, remember that the slope is a ratio that describes both the direction and steepness of the line—key attributes that define the line's equation and graph.

Coordinate Geometry

In the realm of coordinate geometry, every point on the Cartesian plane is defined by a pair of numerical coordinates indicating its position in relation to two perpendicular axes, usually labeled as the X (horizontal) and Y (vertical) axes. A fundamental aspect of coordinate geometry is the concept of the slope, which tells us how a line behaves in space. The exercise involving points A \( (-2, 1) \) and B \( (2, 2) \) gives us a tangible example of how coordinate geometry functions. By determining the slope of the line connecting these two points, we not only understand the line's steepness and direction but also gain insight into the relationship between distinct points in a plane. Students should note that coordinate geometry enables us to calculate distances, midpoints, slopes, and much more, serving as an essential tool for solving geometric problems on a plane.

Undefined Slope

When a line is vertical, meaning it runs parallel to the Y-axis, we say it has an undefined slope. This is because the formula for slope, which involves dividing by the difference in X-coordinates, results in a division by zero when applied to vertical lines. For vertical lines, the X-coordinates of any two points on the line are identical, leading to \( x_2 - x_1 = 0 \). Since division by zero is undefined in mathematics, the slope of a vertical line is likewise undefined. An easy way to spot a vertical line is through the consistency of its X-coordinates. As in the exercise, if we had points A and B with the same X-coordinate but different Y-coordinates, we would have an undefined slope. However, in this particular case, the differing X-coordinates of points A and B led to a well-defined, positive slope.

Direction of a Line

The direction of a line is a visual representation of whether a line rises or falls as it moves from left to right. This concept is directly tied to the sign of the slope. A positive slope, such as \( m = \frac{1}{4} \) found in our example, implies that the line rises; it moves upward as one looks from left to right. Conversely, a negative slope would indicate that the line falls. If the slope is zero, which occurs when the Y-coordinates of two points are the same, the line is horizontal. Understanding the direction of a line aids in graphing linear equations and can even provide insights into real-world trends, like stock prices over time. In our case, knowing that the line rises, we can visualize the line of the graph sloping upwards between points A \( (-2, 1) \) and B \( (2, 2) \).