Problem 10 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((4,2)\) and \((3,4)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the given points

The given points are (4,2) and (3,4). Let's denote them as (x1,y1) = (4,2) and (x2,y2) = (3,4).

Step 2: Apply the slope formula

Use the formula for the slope of a line, which is (y2 - y1) / (x2 - x1). Here, that would be (4 - 2) / (3 - 4) = 2 / -1 = -2.

Step 3: Determine the nature of the line

Since the computed slope is negative (-2), it indicates the line is falling when moving from left to right on a graph.

Key Concepts

Slope Formula

Understanding the slope formula is crucial for analyzing the direction and steepness of a line on a graph. The formula \( m = \frac{y2 - y1}{x2 - x1} \) is used to calculate the slope \(m\) where \((x1, y1)\) and \((x2, y2)\) are coordinates of any two distinct points on the line. To illustrate, let's consider the points \((4,2)\) and \((3,4)\).
When applying the formula, we subtract the y-value of the first point from the y-value of the second point and divide the result by the subtraction of the first point's x-value from the second point's x-value: \(m = \frac{4 - 2}{3 - 4} = -2\).
A slope of \(-2\) tells us two essential things: for every one unit you move to the right (along the x-axis), the line falls two units (along the y-axis). Recognizing these movements can help decipher not just the slope but also the overall behavior of linear functions in various mathematical contexts.

Line Graphs

The line graph is a visual representation of the relationship between two variables. It's a way to depict the function of a line on an X-Y coordinate system. For students working with line graphs, remember that the graph extends in both directions without end, even though we only see a portion of it.
In our example, plotting the points \((4,2)\) and \((3,4)\) and drawing the line through them will show how it slopes downwards from left to right. This graphical representation not only confirms the slope calculation but also provides a visual indication of the line's rate of change—an essential aspect of understanding lines in algebra and geometry.
When studying line graphs, it is also vital to consider axes intercepts, curvature (which should be none for linear equations), and whether the line is continuous or has breaks, although in the case of linear functions, the graph should always be a straight line.

Undefined Slope

Occasionally, you may encounter a line with an undefined slope. This arises with vertical lines, where all the points on the line have the same x-value but different y-values. For instance, if a line passes through the points \((2,3)\) and \((2,7)\), trying to calculate the slope using the formula risults in a division by zero, which is undefined in mathematics.
Why does this happen? The slope represents the ratio of the vertical change to the horizontal change between any two points on the line. Hence, with no horizontal change (since \(x2 - x1 = 0\)), the slope cannot be determined. In our initial example, there is a valid slope of \(-2\), indicating that we're not dealing with a vertical line. However, should students ever come across a vertical line scenario, they should remember that these lines are perfectly upright, and therefore the concept of 'rise over run' is not applicable. In such cases, rather than trying to assign a numerical value, we simply classify the slope as undefined.