Problem 40 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \( (-1.75, -8.3) \), \( (2.25, -2.6) \)

Step-by-Step Solution

Answer

The slope of the line passing through the points \( (-1.75, -8.3) \) and \( (2.25, -2.6) \) is 1.425.

Step 1: Plot the points

The points given are \( (-1.75, -8.3) \) and \( (2.25, -2.6) \). Plot these points on a graph. The first number in each pair is the 'x' coordinate, and the second number is the 'y' coordinate. So, the point \( (-1.75, -8.3) \) is located at the position -1.75 on the x-axis and -8.3 on the y-axis. Similarly, the point \( (2.25, -2.6) \) is located at position 2.25 on the x-axis and at -2.6 on the y-axis.

Step 2: Use the slope formula

The formula for finding the slope between two points \( (x1, y1) \) and \( (x2, y2) \) is \( (y2 - y1) / (x2 - x1) \). Plug in the x and y coordinates of the two points into this formula.

Step 3: Calculate the slope

Plug the coordinates of the points into the formula: \( (-2.6 + 8.3) / (2.25 + 1.75) = 5.7 / 4 = 1.425 \). So, the slope of the line passing through the points \( (-1.75, -8.3) \) and \( (2.25, -2.6) \) is 1.425.

Key Concepts

Plotting Points

Understanding how to plot points on a coordinate plane is essential in graphing a line, which is the visual representation of a linear equation. The coordinate plane has two axes: the horizontal axis, known as the x-axis, and the vertical axis, called the y-axis. Each point on the plane is defined by an ordered pair \( (x, y) \), where 'x' represents the position along the x-axis and 'y' represents the position along the y-axis.

When plotting the points \( (-1.75, -8.3) \) and \( (2.25, -2.6) \), you start by locating the x-coordinate on the x-axis, then move vertically to the y-coordinate. For example, for \( (-1.75, -8.3) \), you would find -1.75 on the x-axis and move down to -8.3 on the y-axis, placing a dot precisely at that intersection. Similarly, for the point \( (2.25, -2.6) \), you find 2.25 on the x-axis and move down to place a dot at -2.6 on the y-axis. Connecting these two points will give you a visual of the line they form.

Slope Formula

The slope of a line measures its steepness and direction. To calculate the slope, you can use the slope formula \( m = \frac{y2 - y1}{x2 - x1} \), where \( (x1, y1) \) and \( (x2, y2) \) are any two distinct points on the line. Think of \(m\) as the vertical change (rise) over the horizontal change (run) between the two points.

To illustrate this with the given points, \( (-1.75, -8.3) \) as \( (x1, y1) \) and \( (2.25, -2.6) \) as \( (x2, y2) \) gives us \( m = \frac{-2.6 - (-8.3)}{2.25 - (-1.75)} \). Simplifying this fraction will yield the line's slope. Remember, a positive slope indicates the line ascends from left to right, while a negative slope means it descends. Moreover, the larger the absolute value of the slope, the steeper the line.

Slope of a Line

The slope is a crucial concept in understanding the behavior of a line in coordinate geometry. The slope indicates both the direction and the steepness of the line. Lines with positive slopes rise from left to right, whereas lines with negative slopes fall. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

Using the given exercise, the calculated slope of 1.425 suggests that for each unit you move to the right along the x-axis, the line goes up approximately 1.425 units. This slope is an expression of how quickly y increases as x increases. By comprehending the slope, you can make predictions about the line: how two different lines might intersect or whether they are parallel (same slope) or perpendicular (slopes are negative reciprocals of each other).