Problem 34 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \((-2, 1)\), \((-4, -5)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (-2, 1) and (-4, -5) is 3.

Step 1: Plotting the Points

Start by plotting the points (-2, 1) and (-4, -5) on a coordinate system. These coordinates represent the positions (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The first point (-2, 1) is located 2 units to the left and 1 unit up from the origin. The second point (-4, -5) is located 4 units to the left and 5 units down from the origin.

Step 2: Drawing the Line

Draw a line that passes through these two points. This line is the visual representation of all points that fall between (-2, 1) and (-4, -5).

Step 3: Finding the Slope

The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the following formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plug the values of the points into the formula:\[ m = \frac{-5 - 1}{-4 - (-2)} = \frac{-6}{-2}\].

Step 4: Simplifying the Slope

After calculation, we find that the slope \( m = 3 \), which is the final answer.

Key Concepts

Graphing points

Graphing points on a coordinate plane is like placing markers on a map. Each point gets a pair of numbers known as coordinates, specifically \(x, y\). The first number, \(x\), tells how far to move horizontally from the origin, and the second number, \(y\), shows how far to move vertically.
For example, consider the point \((-2, 1)\):

• The \(-2\) means you start at the origin and go 2 units to the left.
• The \(1\) indicates moving 1 unit up from your last position.

Overall, plotting points is about following these steps for each coordinate pair you have. It's essential for understanding where lines or curves will go when drawn on the plane.

Coordinate plane

A coordinate plane is a two-dimensional surface formed by a horizontal line called the x-axis and a vertical line called the y-axis. These two lines intersect at a point called the origin, which is the center of all activity on the plane.
Think of the coordinate plane as a large piece of graph paper with a set of invisible lines. They help you navigate where each point or object should lie:

• The x-axis tells how far left or right a point is.
• The y-axis determines how high or low a point is.

Every location on this plane can be described by coordinates \(x, y\), allowing you to precisely map out graphs, plots, or lines.

Equation of a line

An equation of a line expresses a straight path on the coordinate plane and is usually written in the slope-intercept form: \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.
The slope, \(m\), tells how steep the line is. If it is positive, the line rises as it moves from left to right. If it is negative, it falls. In our example, the slope was calculated to be 3, indicating a steep upward line.

• Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) allows you to calculate how flat or steep your line will be.
• Y-intercept \(b\) tells where your line will meet the y-axis, making it a key reference point.

Using the equation of a line, one can draw the line with accuracy and understand the relationship between the coordinates in a more comprehensive way.