Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. $$(-6,-1),(-6,4)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the coordinates

The coordinates are (-6,-1) and (-6,4), which can be written as \(x_1 = -6\), \(y_1 = -1\), \(x_2 = -6\), and \(y_2 = 4\).

Step 2: Calculate the slope

Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{-6 - (-6)}\), Because the denominator is 0, the slope is undefined. This corresponds to a vertical line.

Step 3: Graphing the line

Using a graphing tool, plot the points (-6,-1) and (-6,4) and draw a vertical line passing through them. This line represents the solution.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry that expresses the steepness or incline of a line. It is represented by the letter \( m \). To find the slope when given two points, \((x_1, y_1)\) and \((x_2, y_2)\), you can use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the change in \( y \) (vertical change) over the change in \( x \) (horizontal change). If both changes are non-zero, you will get a defined value that indicates how much \( y \) increases or decreases as \( x \) increases by 1 unit. Remember:

• Positive slope: line rises from left to right.
• Negative slope: line falls from left to right.
• Zero slope: line is perfectly horizontal.

The slope can also be undefined if the line is vertical, which leads us into our next concept.

Vertical Line

A vertical line in coordinate geometry occurs when two points share the same \( x \)-coordinate. In such cases, the formula for the slope becomes:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_2 - y_1}{0} \]Since division by zero is undefined, the slope of a vertical line is also undefined. This implies that vertical lines cannot be expressed by a standard slope value.
Characteristics of vertical lines include:

• Infinite slope.
• No consistent rate of change for \( y \) relative to \( x \).
• Graphically represented as a straight line going up and down.

In the original exercise, the line passing through the points \((-6, -1)\) and \((-6, 4)\) is vertical because both share the same \(x\)-coordinate.

Point Plotting

Point plotting is the process of marking specific locations on a graph based on coordinate pairs. Each pair, \((x, y)\), determines a unique point.
Steps for Point Plotting:

• Identify the \( x \)-coordinate, move horizontally from the origin.
• Locate the \( y \)-coordinate, travel vertically from the \( x \).
• Mark the intersection of \(x\) and \(y\) on the graph.

Ensure accuracy by checking both coordinates before marking the point. In the exercise, the points \((-6, -1)\) and \((-6, 4)\) are plotted by moving 6 units left from the origin and placing a mark at the respective \(y\)-value.

Graphing Utility

A graphing utility is a software tool or calculator function used for plotting points, drawing lines, and visualizing equations in graphical form. It streamlines the process of visualizing mathematical concepts by automating manual graph plotting tasks.
Benefits of Using a Graphing Utility:

• Quickly plots points and draws lines for better understanding.
• Allows for exploration and visualization of various math equations.
• Helps confirm calculations visually and makes adjustments easier.

The exercise involved using a graphing utility to plot points and draw the vertical line connecting them, demonstrating this tool's practical application in solving and visualizing coordinate geometry problems.