Problem 31 - College Algebra

Question

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line). $$ (-1,2)(0,4)(2,1) $$

Step-by-Step Solution

Answer

The points are not collinear; their slopes are different.

Step 1: Understand the Given Points

We are given three points: \((-1, 2)\), \((0, 4)\), and \((2, 1)\). These points are in the form of \((x, y)\) coordinates on a 2D Cartesian coordinate plane. Our task is to determine if these points are collinear, which means they lie on the same straight line.

Step 2: Plot the Points on a Coordinate Plane

To begin, plot each point on a coordinate plane:- Point A is \((-1, 2)\): start at the origin (0,0), move left 1 unit, and move up 2 units to plot.- Point B is \((0, 4)\): start at the origin, do not move horizontally, and move up 4 units to plot.- Point C is \((2, 1)\): start at the origin, move right 2 units, and move up 1 unit to plot.

Step 3: Analyze Collinearity Using Slope

Calculate the slope between each pair of points:- Slope of line AB (\((-1, 2)\) to \((0, 4)\)) is \(\frac{4-2}{0+1} = 2\).- Slope of line BC (\((0, 4)\) to \((2, 1)\)) is \(\frac{1-4}{2-0} = -\frac{3}{2}\).- Slope of line AC (\((-1, 2)\) to \((2, 1)\)) is \(\frac{1-2}{2+1} = -\frac{1}{3}\).Since the slopes are not equal, the points are not collinear.

Key Concepts

Cartesian Coordinates

In coordinate geometry, the Cartesian coordinate system is a 2D plane defined by two perpendicular axes: the x-axis and the y-axis. Points are identified on this plane using pairs called coordinates, written as \((x, y)\). The x-coordinate specifies horizontal movement from the origin, while the y-coordinate specifies vertical movement.

For instance, the point \((-1, 2)\) means you move 1 unit to the left along the x-axis (since it is negative) and 2 units up along the y-axis. The origin is at the very center of the coordinate plane, marked as (0,0). Every point has a unique position on this plane based on its coordinates.

Understanding Cartesian coordinates is crucial because it helps visualize mathematical concepts and is used in various applications from navigation to computer graphics.

Collinearity

Collinearity is a term used to describe points that lie on the same straight line. When three or more points are collinear, you can draw a single straight line through all the points. This concept is vital in mathematics as it helps to understand linear relationships.

To determine if points are collinear, we can use various mathematical methods. The most common is checking their slopes. If the slope (the steepness of the line) between any pair of points remains the same, the points are collinear. In the exercise, however, you discovered that the slopes differ, which indicates that the points \((-1, 2)\), \((0, 4)\), and \((2, 1)\) do not fall on a single line. Thus, they are not collinear.

Recognizing non-collinear points is just as interesting because they form shapes like triangles and polygons, leading to further exploration of geometry.

Slope Calculation

The slope in coordinate geometry is a measure of how steep a line is. It shows the rate at which the y-coordinate changes per unit change in the x-coordinate between two points. The slope formula is given as: \[ \text{Slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} \]

In our exercise, calculating the slopes between each pair of points helped us determine collinearity. For instance, the slope between points \((-1, 2)\) and \((0, 4)\) is calculated as \(\frac{4-2}{0+1} = 2\). It indicates a line that rises 2 units vertically for every 1 unit it moves horizontally.

By comparing the different slopes, it was determined that they differed (2, -\(\frac{3}{2}\), and -\(\frac{1}{3}\)), meaning the points form different lines.

Understanding slope calculation is essential for graphing lines and determining the relationship between variables in algebra and calculus.

Plotting Points

Plotting points is the fundamental action of placing a point on the Cartesian plane based on its coordinates. It involves a few simple steps which ensure accuracy and proper understanding of graphing.

To plot a point correctly:

• Start at the origin (0,0).
• Move horizontally to match the x-coordinate. For negative x-values, move left; for positive x-values, move right.
• Move vertically to match the y-coordinate. For positive y-values, move up; for negative y-values, move down.

As an example, the point \( (2,1) \) is plotted by moving 2 units to the right of the origin and 1 unit up. Accurate plotting allows for analyzing spatial relationships, such as the collinearity explored in the exercise, and is the basis for learning functions and graph interpretation in higher mathematics.