Problem 5 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(5-8,\) let \(L\) be the line determined by points \(A\) and \(B .\) \(\begin{array}{ll}{\text { (a) Plot } A \text { and } B .} & {\text { (b) Find the slope of } L} \\ {\text { (c) Draw the graph of } L .}\end{array}\) $$A(1,-2), \quad B(2,1)$$

Step-by-Step Solution

Answer

The slope of the line L is 3.

Step 1: Plotting Points A and B

Plot the points \( A(1,-2) \) and \( B(2,1) \) on the graph. This can be done by marking an x value of 1 and y value of -2 for point A, and an x value of 2 and y value of 1 for point B.

Step 2: Finding the Slope of Line L

The slope, \( m \), of the line through points A and B is calculated by the formula \( m= \frac{y2-y1}{x2-x1} \). By substituting \( x_1 = 1, y_1 = -2, x_2 = 2, y_2 = 1 \), we get \( m = \frac{1-(-2)}{2-1} = 3 \). Hence, the slope of line L is 3.

Step 3: Drawing the Graph of Line L

Using the slope and the points A and B, draw the line L on the graph. The slope of 3 means the line rises by 3 units for each unit it moves to the right. Starting at point A, move one unit right and three units up, this should give point B. Connect points A and B to draw the line L.

Key Concepts

Plotting Points on a Graph

Understanding how to plot points on a graph is a fundamental skill in algebra and provides the basis for visualizing relationships between variables. To plot a point, you need to have a coordinate pair \(x, y\), where \(x\) represents the horizontal distance from the origin and \(y\) the vertical distance. Imagine your graph as a map, with the origin (0,0) being the point where the horizontal axis (x-axis) and the vertical axis (y-axis) intersect.

When plotting a point like \(A(1, -2)\), start at the origin and count 1 unit to the right, as the x-value suggests. Then, from there, go 2 units down, as indicated by the negative y-value. For point \(B(2, 1)\), move 2 units to the right and then 1 unit up. Mark these points on your graph with a clear dot. Correct plotting ensures that all subsequent steps, such as drawing a line through these points, are accurate.

Calculating Slope

The concept of slope is one of the cornerstones of algebra, especially when dealing with linear equations. The slope is a measure of how steep a line is and which direction it goes. Formally, the slope (usually denoted as \(m\)) is the ratio of the change in the y-values to the change in the x-values between two distinct points on the line.

The formula to calculate the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In our example, with points \(A(1, -2)\) and \(B(2, 1)\), we can label \(A\) as \(\left(x_1, y_1\right)\) and \(B\) as \(\left(x_2, y_2\right)\). Substituting the values into the formula gives us \( m = \frac{1 - (-2)}{2 - 1} = 3 \), which means for every 1 unit increase in x, y increases by 3 units. The slope tells us that line L rises quickly and moves to the right, a pattern you can visualize and effectively use when drawing the graph of the line.

Linear Equations

Linear equations form a straight line when graphed and are one of the most basic yet powerful tools in algebra. They follow the general form \(y=mx+b\), where \(m\) is the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis. When graphing a line, knowing the slope and one point on the line is sufficient to draw the entire line.

In the given exercise, using the calculated slope of 3 and one of the points, point A at \(1, -2\), for instance, allows us to draw line L. You would start at point A, and then follow the slope by moving one unit to the right (positive x-direction) and three units up (positive y-direction) to reach another point on the line. Connect these points with a ruler to ensure the line is straight. Remember that a linear equation represents a constant rate of change, which is visually reflected by the uniform slope of the line across the graph.