Problem 6 - Precalculus Essentials

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(4,-1) \text { and }(3,-1)$$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (4,-1) and (3,-1) is 0. The line is horizontal.

Step 1: Identify the Given Points

The points that are given in the exercise are (4,-1) and (3,-1). So, you have \(x1=4, y1=-1, x2=3, y2=-1\)

Step 2: Find the Slope

The slope of the line passing through these points is calculated using the formula \(m=\frac{y2-y1}{x2-x1}\). Plug the coordinates of the points into the formula to get \(m=\frac{-1-(-1)}{3-4}=\frac{0}{-1}=0\). Therefore, the slope is 0.

Step 3: Determine the Line Direction

Based on the calculated slope, which is 0, the direction of the line can be determined. A line with a slope of 0 is a horizontal line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is crucial for graphing and analyzing linear relationships between two variables. The general formula for the slope-intercept form is given by \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis.

In the context of the provided exercise, once the slope ( \( m \) ) is known, if we also have the y-intercept ( \( b \) ), we could easily write the equation of the line in slope-intercept form. For instance, with a slope of 0 and if the y-intercept were known, the equation of the line would simply be \( y = b \).

The benefit of using this form is its directness in helping to graph a line and understand how the line changes as one variable increases or decreases. It shows the rate of change (slope) and the starting value (y-intercept) very clearly.

Undefined Slope

In mathematical terms, a line with an undefined slope is one that runs vertically on a graph. This occurs when the x-coordinates of a pair of points are the same, leading to a division by zero when calculating the slope using the formula \( m = \frac{y2 - y1}{x2 - x1} \).

For example, consider two points \((3,-1)\) and \((3,2)\). Calculating the slope would involve \( \frac{2 - (-1)}{3 - 3} \) which simplifies to \( \frac{3}{0} \), and since division by zero is undefined in mathematics, so is the slope in this case.

This concept is essential because it distinguishes vertical lines from all other types of lines on a graph. When dealing with vertical lines, we can't use the slope-intercept form since there is no finite slope to plug in. Instead, the equation of a vertical line will simply be \( x = \) a constant value.

Horizontal and Vertical Lines

The concepts of horizontal and vertical lines pertain to their orientation in the Cartesian plane. Each type of line has a distinct slope characteristic; horizontal lines have a slope of 0, while vertical lines have an undefined slope.

A horizontal line indicates a constant y-value for all x-values, which is why the slope is 0; there is no rise over run as the formula for slope suggests ( \( m = \frac{rise}{run} \) ). Mathematically, the equation for a horizontal line is given by \( y = \) a constant value.

Conversely, a vertical line represents a constant x-value regardless of the y-value. As discussed earlier, the slope is undefined because you cannot divide by zero. The equation for a vertical line is \( x = \) a constant value.

Understanding these concepts is crucial, especially when analyzing the relationship between two variables, plotting on the coordinate plane, or solving geometric problems. The exercise provided demonstrates a horizontal line which, due to its slope of 0, indicates no vertical change as one moves along the points of the line.