Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((-3,4)\) \(m=0\)

Step-by-Step Solution

Answer

The three additional points through which the line passes are (-2,4), (-1,4) and (0,4)

Step 1: Identify Given Point and Slope

The given point on the line is (-3,4), and the slope m equals 0. This means our line is horizontal and thus, the y-coordinate for any point on this line will be 4.

Step 2: Find Additional Points Using the Slope

As the slope is zero, it means we move neither up nor down. All we need to do to find more points is choose any values for x. Let's choose -2, -1, and 0 for x.

Step 3: Applying X-values

The new points will be (-2,4), (-1,4) and (0,4), consistently maintaining the y-coordinate value of 4.

Key Concepts

Horizontal Line Equation

When working with the equation of a horizontal line in coordinate geometry, you should understand that it has a very specific form. A horizontal line runs left to right and has the same y-coordinate for all points that lie on it.

For any horizontal line, the equation can be written as: \( y = b \) where \( b \) is the y-coordinate of any point on the line. To put it into context using the exercise, a line that passes through the point (-3, 4) and has a slope of 0 is given by the equation \( y = 4 \) because the y-coordinate remains constant across all points on the line.

When you're looking to identify a horizontal line on a graph, just remember that its slope will always be 0, it will be parallel to the x-axis, and the equation does not contain an x variable.

Slope of a Line

The slope of a line is a measure of how steep the line is and is represented by the letter \( m \). It is calculated as the rise over the run, which is the change in y-coordinates divided by the change in x-coordinates between two points on the line.

In a graphical representation, if a line goes up from left to right, it has a positive slope. If it goes down from left to right, it has a negative slope. A horizontal line, like the one in the exercise with a slope of 0, has no rise; it doesn't go up or down as it moves from left to right. As a result, the equation for the slope of any horizontal line is \( m = 0 \).

A vertical line, on the other hand, goes straight up and down and has an undefined slope because you would be dividing by zero (the run is zero) which is not allowed in mathematics. Remember, knowing the slope allows you to predict the direction and steepness of a line on a graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method enables calculations using algebraic equations and can uniquely determine the position of points on a two-dimensional plane using ordered pairs, typically \( (x, y) \) coordinates.

The horizontal line example from the exercise fits neatly into this system. By knowing one point, \( (-3, 4) \), and understanding that the slope of the line \( m = 0 \), you can infer that other points sharing the same y-coordinate, such as \( (-2, 4) \), \( (-1, 4) \), and \( (0, 4) \) also lie on this line. Coordinate geometry is fundamental in connecting algebraic equations with geometric figures to solve various mathematical problems and is widely applied within fields ranging from computer graphics to engineering and physics.