Problem 7 - Algebra 1: Concepts and Skills

Question

Plot the points and draw the line that passes through them. Without finding the slope, determine whether the slope is positive, negative, zero, or undefined. $$ (-3,1) \text { and }(1,-3) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (-3,1) and (1,-3) is negative.

Step 1: Plotting the Points

Start by drawing an x-y axis on a graph. Next, mark the two given points: (-3, 1) and (1, -3). The first number in a pair is the x-coordinate (horizontal position) and the second one is the y-coordinate (vertical position). Therefore (-3,1) lies on the graph 3 units to the left of y-axis and 1 unit above the x-axis. Point (1, -3) is located 1 unit to the right of the y-axis and 3 units below the x-axis.

Step 2: Drawing the Line

Starting from one point, draw a straight line that goes through both points. The line should continue beyond the points to represent that it keeps on going indefinitely.

Step 3: Determining the Slope

Without calculating the exact slope, it can be determined from the graph's shape kind. From left to right, the line goes down, it means that the slope is negative.

Key Concepts

Plotting Points

Plotting points on a graph is the foundation of graphing equations. Each point is defined by a pair of numbers known as coordinates. These coordinates specify the horizontal and vertical positions on the graph. The first number is the x-coordinate, which tells us how far along the horizontal axis (or x-axis) the point is. The second number is the y-coordinate, which shows the distance along the vertical axis (or y-axis).
This process involves a few simple steps:

• Start by identifying the x-coordinate and move that many units right (if positive) or left (if negative) from the origin of your graph.
• Then, from this x-coordinate location, move perpendicular to the previous line: up the plot for a positive y value and downwards for a negative y value.
• Place the point where you end up.

By accurately plotting the points given, you create a visual representation of their positions on the graph.

Graphing Linear Equations

Once the points are plotted, the next step is connecting them with a straight line. This line represents the solution set for a linear equation containing two variables. In our case, we have the points (-3,1) and (1,-3).
To draw the line:

• Locate your first plotted point and start the line from there.
• Then, aim the line towards your second point, ensuring that it smoothly passes through that location too.
• Extend the line beyond both points on either side to represent continuity.

This process forms the graph of the linear equation. The presence of just a single straight line verifies that there is a constant rate of change between the x and y variables, a signature characteristic of linear equations.

X-Y Coordinate System

Understanding the x-y coordinate system is crucial for graphing efficiently. It serves as the framework for plotting points and graphing equations. The x-axis is the horizontal line that runs left and right, and the y-axis is the vertical line that goes up and down. The point where the x and y axes intersect is called the origin, with coordinates (0,0).
This system allows us to map pairs of numbers onto a graph making it easier to visualize numeric relationships. Here's how it is beneficial:

• Organizes numerical data into a visual format, making analysis simpler.
• Facilitates the identification and interpretation of patterns and trends.
• Allows for precise plotting which helps in finding solutions to mathematical equations.

The convenience of the x-y coordinate system lies in its ability to handle complex data visually, making sense of mathematical equations, functions, and the relationships they describe.