Problem 66 - Calculus Volume 1

Question

For each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. . (1,4) and (1,0)

Step-by-Step Solution

Answer

The line is vertical with an undefined slope.

Step 1: Understanding Slope Formula

The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(m\) is the slope.

Step 2: Plug Points into Slope Formula

Substitute the given points \((1,4)\) and \((1,0)\) into the slope formula: \[ m = \frac{0 - 4}{1 - 1} \]

Step 3: Calculate the Denominator

The denominator is calculated as \(1 - 1 = 0\).

Step 4: Division by Zero

Since the denominator is zero, the slope formula yields division by zero, indicating that the line is vertical.

Step 5: Determine Line's Direction

For vertical lines, the slope is undefined, and the line is classified as vertical.

Key Concepts

Vertical Lines

Imagine skyscrapers reaching up into the sky; that's essentially what vertical lines do on a graph! A vertical line goes straight up and down. In our example, when you look at the points (1, 4) and (1, 0), they share the same x-coordinate of 1. This is a key indicator of a vertical line.

Vertical lines are unique because they do not run from left to right but rather top to bottom. This is why, when viewing a vertical line, there's no horizontal movement between points. It's purely vertical, like an elevator ride. This characteristic forms the basis of why vertical lines have no defined slope. Without any horizontal change, there's no way to calculate a rate of ascent or descent—hence, an undefined slope!

In essence, the x-value remains constant, showcasing the line's defining vertical characteristic.

Slope Formula

The slope formula is an easy way to determine how steep a line is or its direction. Think of it as a set of instructions for identifying the slant of a path on a map.

Using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] we subtract the y-coordinates and then the x-coordinates of any two points on the line. The result, \(m\), tells us the slope.

Here’s a buddy tip: when the slope \(m\) is positive, the line inclines upwards, like a mountain hike. If it’s negative, it descends, like going downhill. For horizontal lines, the slope is zero because there's no vertical change.

However, bear in mind that when the denominator becomes zero (like in our previous example), this can lead to an important exception: the slope won't be defined.

Division by Zero

Division by zero isn't something you can do, just like you can't divide a pizza by no people—it simply doesn't make sense!

When using the slope formula, and the denominator—\(x_2 - x_1\)—turns out to be zero, you face the division by zero conundrum. In our example with points (1, 4) and (1, 0), the denominator is 0, which indicates no horizontal spacing between the points.

Division by zero has no mathematical solution, meaning it's undefined. This is how we spot vertical lines: the zero denominator tells us the line shoots straight up!

Always remember, you cannot perform division by zero in math problems, and it creates a signal for vertical alignment in terms of graphing lines.