Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Plot the points and find the slope of the line passing through the points. \((-6,-1),(-6,4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((-6,-1)\) and \((-6,4)\) is undefined.

Step 1: Plot the points

Plot the points \((-6,-1)\) and \((-6,4)\) on a graph. Both points fall on the vertical line where x=-6.

Step 2: Calculate the slope

To calculate the slope of the line passing through these points, use the formula \[slope = \frac{y2 - y1}{x2 - x1}\]. Here, \(x1 = x2 = -6\), \(y1 = -1\), and \(y2 = 4\). Therefore, the slope is \[slope = \frac{4-(-1)}{-6-(-6)} = \frac{5}{0}\]. However, division by 0 is undefined.

Step 3: Evaluate the slope

Since the division by 0 is undefined, this indicates that the slope of the line passing through the points is undefined. This is consistent with the line being vertical.

Key Concepts

Graphing Points

Graphing points is the first step in understanding the behavior and properties of lines on a coordinate plane. Each point on a graph is defined by a pair of coordinates, \(x, y\), indicating its position relative to the horizontal x-axis and the vertical y-axis.
To graph a point, locate the x-coordinate along the x-axis, then move vertically to the location specified by the y-coordinate.

• For \((-6,-1)\), move to -6 on the x-axis, then drop down to -1 on the y-axis.
• For \((-6,4)\), move to -6 on the x-axis, then rise up to 4 on the y-axis.

Placing these points on the graph reveals that they lie directly above each other on the same vertical line where x equals -6. This demonstrates one key feature of vertical lines: all points have the same x-coordinate.

Undefined Slope

The concept of slope gives us insight into how steep a line is and its direction. The slope is found using the formula \[\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.\]\
It measures the change in the y-coordinates (vertical change) over the change in the x-coordinates (horizontal change).
However, when graphing points like \((-6,-1)\) and \((-6,4)\), both x-coordinates are the same, making \(x_2 - x_1\) equal to zero. Any time we attempt to divide by zero, the result is undefined.

• For these points: \(\frac{4 - (-1)}{-6 - (-6)} = \frac{5}{0}\).

Because division by zero is undefined, the slope itself is undefined. This helps us characterize lines that are vertical.

Vertical Line

Vertical lines are a specific type of line on a coordinate plane where all points share the same x-coordinate. This makes them stand straight up and down, perpendicular to the x-axis.
Vertical lines have several important characteristics:

• The equation of a vertical line is always in the form \(x = c\), where c is a constant.
• Vertical lines have an undefined slope because there's no change in the x-value as you move along the line.
• When plotting on a graph, vertical lines will not cross the x-axis but will continue parallel to it.

In our example, the line with points \((-6,-1)\) and \((-6,4)\) forms such a vertical line. It perfectly demonstrates the characteristics of having no horizontal change and an undefined slope.