Problem 13 - 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. $$(7,4),(-1,8)$$

Step-by-Step Solution

Answer

The slope of the line is positive because the line rises from left to right.

Step 1: Plot the Points

Start by drawing a Cartesian plane to accommodate our points (7,4) and (-1,8). Place these points, using the respective x and y for each point.

Step 2: Draw a line through the points

After plotting the points, draw a line that passes through both points.

Step 3: Determine the Slope

Without calculating, the slope of the line can be determined by observation. If the line rises from left to right, then the slope is positive. If it falls from left to right, the slope is negative. If the line is horizontal, then the slope is zero, and if the line is vertical, it is undefined. In this case, the line slopes upward from left to right, so the slope is positive.

Key Concepts

Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system used in mathematics to locate points and graph lines or curves. It consists of two perpendicular axes: the horizontal axis (called the x-axis) and the vertical axis (called the y-axis).
These axes intersect at the origin, which is represented by the coordinates (0,0). Together, they create four quadrants that help us specify the location of any point.

• The first quadrant is where both x and y values are positive.
• The second quadrant has negative x values and positive y values.
• The third quadrant includes points where both x and y are negative.
• The fourth quadrant contains points with positive x values and negative y values.

Using this system, any point on the plane can be represented by an ordered pair \((x, y)\), where \(x\) is the point's horizontal position and \(y\) is its vertical position.

Plotting Points

Plotting points on the Cartesian plane is a method to represent specific locations graphically. Each point is defined by its "coordinates," which come in the form of an ordered pair \((x, y)\).
To plot a point:

• Start from the origin (0,0).
• Move horizontally along the x-axis to the x-coordinate of the point.
• From that position, move vertically to the y-coordinate of the point.

For example, plotting the point (7,4) involves moving 7 units right along the x-axis from the origin and then moving 4 units up parallel to the y-axis. Similarly, to plot the point (-1,8), you would move 1 unit left (because -1 is negative) and then climb 8 units upward. Once you have placed the points, you can connect them using a straight line.

Positive and Negative Slope

The slope of a line is a measure of its steepness and direction on the Cartesian plane. This concept helps determine how a line inclines or declines.
Without complex calculations, you can ascertain the slope by observing the line's trajectory:

• If the line ascends from the left to the right, then it has a **positive slope.**
• A line descending from left to right indicates a **negative slope.**
• If the line remains perfectly horizontal, its slope is **zero.**
• A perfectly vertical line has an **undefined slope** because it doesn't run parallel with the x-axis.

In the original exercise, plotting the points (7,4) and (-1,8) and then drawing a line that connects them demonstrates an upward trend from left to right. This visual proof tells us that the line has a positive slope.