Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Plot the points and find the slope of the line passing through the points. \((2,4),(4,-4)\)

Step-by-Step Solution

Answer

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

Step 1: Plot Points

Plot the given points (2,4) and (4,-4) on a two-dimensional Cartesian plane.

Step 2: Formula for Slope

Understand that the slope (m) of a line passing through two points can be calculated using the formula m = \((y_2 - y_1) / (x_2 - x_1)\).

Step 3: Substitute Values

Substitute the given points into the slope formula. Where (x1,y1) = (2,4) and (x2,y2) = (4,-4). So, substituting these values, the slope formula becomes m = \((-4 - 4) / (4 - 2)\)

Step 4: Calculate Slope

Calculate the difference \(y_2 - y_1\) and \(x_2 - x_1\) to find the slope. m = -8/2 = -4.

Key Concepts

Cartesian Plane

The Cartesian plane is a vital element in mathematics. It offers a clear way to visualize algebraic equations and their solutions. Imagine it as a graph with two intersecting lines. One line goes horizontally and is called the x-axis. The other line goes vertically and is called the y-axis. The point where these two axes meet is known as the origin. It is represented by the coordinates (0, 0). Each point on this plane is an ordered pair of numbers known as coordinate points.

The horizontal position of a point is determined by its x-coordinate, and the vertical position by its y-coordinate. This system helps in plotting points and seeing how they relate to each other. Whenever you are calculating the slope, you are essentially looking at how a line behaves on this plane. Lines can go up, down, or remain flat. The Cartesian plane is instrumental in exploring these relationships visually.

Slope Formula

When dealing with two points on a Cartesian plane, the slope of a line is a key concept. It tells you how steep the line is, and in which direction it goes. The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here,

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

The slope, represented by \(m\), is simply the change in the y-values divided by the change in the x-values. This is frequently described as 'rise over run.' This formula helps in understanding how a line tilts in the Cartesian plane.

A positive slope means the line goes up as it moves from left to right. A negative slope means it goes down. Zero slope suggests a flat line. This formula is foundational for many algebraic concepts and helps in understanding lines and angles.

Coordinate Points

Coordinate points are like addresses on the Cartesian plane. They help locate any position on the graph precisely. Each point consists of an x-coordinate and a y-coordinate, represented as an ordered pair \((x, y)\). The x-coordinate tells you how far a point is along the horizontal axis; the y-coordinate shows how far along the vertical axis.

Let's take the points

• \((2, 4)\)
• \((4, -4)\)

as examples. The first point is 2 units to the right of the origin and 4 units up. The second point is 4 units to the right and 4 units down. Each point is crucial when calculating the slope of a line passing through them.

Accurate knowledge about coordinate points is important as they are used in many mathematical calculations, from basic geometry to more advanced topics like trigonometry and calculus. They provide a foundation for plotting and studying the relationships between different mathematical entities on a graph.