Problem 18 - Algebra 1

Question

PIot the points and draw a line through them. Without calculating, state whether the slope of the line is positive, negative, zero, or undefined. Explain your reasoning. $$(1,3),(-2,1)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (1,3) and (-2,1) is positive.

Step 1: Plot the Points

Start by creating a graph and plot the points (1,3) and (-2,1). To do this, for each point, the first number is the x-coordinate and the second is the y-coordinate. So, the point (1,3) would be one unit to the right and three units up from the origin (0,0), and the point (-2,1) would be two units to the left and one unit up from the origin.

Step 2: Draw a Line

After marking the two points, draw a straight line passing through these two points.

Step 3: Analyze the Slope

Once the line is drawn, observe its direction. The point (-2,1) is lower and to the left of the point (1,3). Thus, connecting these points from left to right results in a line that rises. This means the slope of the line is positive.

Step 4: Conclusion

Conclude that the slope of the line passing through the points (1,3) and (-2,1) is positive without calculation, based on the visual analysis of the graph.

Key Concepts

Graphing Points

Graphing points is a fundamental skill that is vital for understanding how lines and slopes work. When you graph points, it's like plotting each dot in a connect-the-dots puzzle. Every point on a graph has its own coordinate pair, usually written as \((x, y)\).

• The first number, \(x\), tells you how far to move horizontally from the origin (\((0, 0)\)), which is the center of your graph.
• The second number, \(y\), tells you how far to move vertically.

In the exercise above, to graph the point \((1,3)\), move one unit right and three units up from the origin. For the point \((-2,1)\), move two units left and one unit up. Once both points are plotted on the graph, you'll have a clear visual reference for drawing a line between them.

Line Direction

The direction of a line on a graph reveals much about its slope. After plotting the points, drawing a line through them helps us determine the slope's characteristics.

• A line that moves from lower left to upper right has an upward or positive direction.
• A line from upper left to lower right moves downward, indicating a negative direction.

In our exercise, plotting points \((1,3)\) and \((-2,1)\) results in a line that moves upward from left to right. This upward line direction means that the slope will be positive. Visualizing line direction at a glance lets you understand the whole concept of slope intuitively, without even calculating numbers.

Positive Slope

A positive slope is an important concept in graphing lines. The slope tells us how steep a line is and in which direction it tilts as we move along the \(x\)-axis.

• When the line rises as it moves from left to right, the slope is positive.
• A greater positive slope means a steeper climb.

In this exercise, as we move from the left point \((-2, 1)\) to the right point \((1, 3)\), the line ascends upward. This results in a positive slope. Recognizing a positive slope helps in predicting how variables will behave in real-world situations, such as determining the speed and acceleration of an object.