Problem 76 - Algebra 1

Question

Plot the points in the table. Determine whether the slope of the line given by the points is positive or negative. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & 7 & 5 & 3 & 1 & -1 & -3 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The plotted line has a negative slope, meaning as x increases, y decreases.

Step 1: Plotting the points

According to the table, The points to plot would be (-2,7), (-1,5), (0,3), (1,1), (2,-1) and (3,-3).

Step 2: Drawing the line

After plotting the points on the graph, connect them with a straight line. Make sure the line passes through each dot.

Step 3: Determining the slope

Observe the direction of the line. If it is going from left to right and moving downwards, the slope is negative. If the line was moving upwards from left to right, the slope would be positive.

Key Concepts

Plotting Points

Plotting points on a coordinate plane is the foundation of graphing any equation. Begin by understanding the coordinate plane, which consists of two perpendicular number lines: the horizontal 'x-axis' and the vertical 'y-axis'. Each point on this plane is defined by an ordered pair of numbers, \( (x, y) \), representing its position in relation to these axes.

To plot a point, move along the x-axis to the 'x' value of the ordered pair and then move vertically to the 'y' value. For example, to plot the point \( (-2, 7) \), start at the origin (where the x and y axes intersect at \( (0, 0) \)), move 2 units to the left (since -2 is negative), and then rise 7 units upward. Mark this position with a dot. Repeat this process for each point given in the table. After plotting all points, you can observe the formation of a specific shape or pattern, which in this case will hint towards a straight line.

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the equation and then connecting these points with a straight line. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis.

After plotting the given points from the exercise on the graph, you will notice that these points align to form a straight line. Ensure your line is accurately drawn through all of the points; use a ruler for precision. This line visually represents the solution set of the linear equation that corresponds to the set of points. Through graphing, not only do you visualize relationships between variables but also understand the concept of slope and intercepts much better.

Negative Slope

The slope of a line measures its steepness and the direction it inclines or declines. When a line on a graph moves from left to right and also moves downwards, it has a negative slope. This indicates that as the 'x' value increases, the 'y' value decreases. In mathematical terms, a negative slope will have a negative 'm' value in the linear equation \( y = mx + b \).

The set of points from the exercise, when plotted and connected, clearly demonstrate a downward trend from left to right, revealing a negative slope. This is essential for understanding the rate at which 'y' decreases relative to 'x'. It is also helpful in predicting future behavior within the context of the represented phenomenon, like a declining temperature over time.