Problem 5 - 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. $$(4,-2) \text { and }(3,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (4,-2) and (3,-2) is 0. Therefore, the line is horizontal.

Step 1: Identify the coordinates

Identify the coordinates from the problem. Here, the two points are (4,-2) and (3,-2). Let's denote them as \((x_1, y_1)\) and \((x_2, y_2)\), respectively. So, \(x_1 = 4\), \(y_1 = -2\), \(x_2 = 3\), and \(y_2 = -2\).

Step 2: Plug into slope formula

Plug these points into the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given values we get \( m = \frac{-2 - (-2)}{3 - 4} = 0 \).

Step 3: Interpret the slope

The slope value 0 means the line is horizontal.

Key Concepts

Horizontal Line

When we talk about a horizontal line in coordinate geometry, we're referring to a line that has a constant y-value across different x-values. This means the line doesn't rise or fall as it moves from left to right; it stays flat. For example, in the exercise, both points

• (4,-2)
• (3,-2)

share the same y-coordinate, which is -2. This tells us that the line connecting these points is horizontal.
The slope of a horizontal line is always 0. You can think of it as the line having no steepness. So, if you picture rolling a ball on it, the ball wouldn't roll up or down because the line isn't tilted.

Undefined Slope

An undefined slope occurs when a line is vertical. In coordinate geometry, a vertical line goes straight up and down, which means it has a constant x-value across different y-values. We can't calculate the slope of a vertical line using the regular slope formula because it involves division by zero, which is undefined in mathematics.
For example, if you have two points like (3,5) and (3,1), they share the same x-coordinate. Plugging these into the slope formula would give:\[m = \frac{1 - 5}{3 - 3} = \frac{-4}{0}\]Since division by zero is not possible, we say the slope is undefined. Vertical lines don't have a numerical slope value, and this is a unique property of them.

Slope Formula

The slope formula is a key tool in coordinate geometry that helps us understand how steep a line is between two points on a graph. The slope is often denoted by the letter "m" and calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where:

• \(x_1, y_1\) are the coordinates of the first point.
• \(x_2, y_2\) are the coordinates of the second point.

The numerator \(y_2 - y_1\) indicates the change in y (the rise), and the denominator \(x_2 - x_1\) shows the change in x (the run). This helps in knowing whether the line is rising, falling, horizontal, or vertical depending on the value of \(m\). If the result is:

• Positive: The line rises.
• Negative: The line falls.
• Zero: The line is horizontal.
• Undefined: The line is vertical.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This allows us to find the properties of figures and graph equations representing lines, curves, and shapes. With the use of coordinates

• (x, y)

we can translate geometric problems into algebraic ones and vice versa.
In the Cartesian coordinate system, we use two axes:

• The x-axis, which extends left and right horizontally.
• The y-axis, which extends up and down vertically.

Every point in this system is defined by its position relative to these axes, making it easier to calculate distances, midpoints, and slopes with formulas like the slope formula. Coordinate geometry is a powerful tool that combines algebraic and geometric principles to solve a wide range of mathematical problems.