Problem 6 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (-8,3) \text { and }(-2,3) $$

Step-by-Step Solution

Answer

The slope of the line is 0.

Step 1: Identify the coordinates of the points

The coordinates of the first point are \((-8, 3)\) and the coordinates of the second point are \((-2, 3)\). These are given in the problem statement.

Step 2: Use the slope formula

The slope (\(m\)) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's substitute the coordinates of the points into the formula.

Step 3: Substitute the values into the slope formula

Substitute \((x_1, y_1) = (-8, 3)\) and \((x_2, y_2) = (-2, 3)\) into the slope formula: \[ m = \frac{3 - 3}{-2 + 8} \] Simplify the numerator and denominator.

Step 4: Calculate the slope

The calculation simplifies as follows:\[ m = \frac{3 - 3}{-2 + 8} = \frac{0}{6} = 0 \]Since the numerator is 0, the slope of the line is 0.

Key Concepts

Understanding Points and Coordinates

In geometry, **points** are fundamental objects. A point is essentially a location on a plane, devoid of any size or dimension. **Coordinates** are numbers that describe that location on a two-dimensional plane. The most common way to express it is through Cartesian coordinates, which are written as o (x, y) The letter "x" typically refers to the horizontal position, while "y" refers to the vertical position on a grid. When you see a point like ((-8, 3)), - The first number, -8, denotes its position on the horizontal X-axis (meaning left -8 units from the origin). - The second number, 3, denotes its position on the vertical Y-axis (meaning up 3 units from the origin). Understanding how to read coordinates is crucial to using them in other concepts such as calculating the slope of a line.

Using the Slope Formula

The **slope** of a line is a measure of its steepness and direction. It is calculated based on the coordinates of two points on the line. Slope can tell us how fast a line ascends or descends as we move along it. The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \)is used to find this value. It calculates the vertical change divided by the horizontal change between two points, - \((x_1, y_1)\) and \((x_2, y_2)\).**Example Calculation:**Given points (-8, 3) and (-2, 3), we substitute:- \(y_2 - y_1 = 3 - 3 = 0\)- \(x_2 - x_1 = -2 + 8 = 6\)This makes the slope:\( m = \frac{0}{6} = 0\)Whenever the result in the numerator is zero, the entire fraction is zero, indicating a special type of line.

What is a Horizontal Line?

A **horizontal line** is a straight line that runs from left to right on a graph. These lines have unique properties: - The slope of a horizontal line is always 0, as there is no vertical movement. Therefore, when you calculate the slope with coordinates like (-8, 3) and (-2, 3), the numerator of the slope formula is 0, causing the result to be 0. - All points on a horizontal line share the same y-coordinate (e.g., 3 in our example). This means no matter where you look on the line, its vertical position doesn't change. Recognizing horizontal lines is important, since they signify constant values without changes in height. They graphically represent situations where there's no increase or decrease as you move along the x-axis.