Problem 8 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(5-8,\) let \(L\) be the line determined by points \(A\) and \(B .\) \(\begin{array}{ll}{\text { (a) Plot } A \text { and } B .} & {\text { (b) Find the slope of } L} \\ {\text { (c) Draw the graph of } L .}\end{array}\) $$A(1,2), \quad B(1,-3)$$

Step-by-Step Solution

Answer

Line L, passing through points A(1,2) and B(1, -3), is a vertical line. Its slope is undefined.

Step 1: Drawing Points A and B

In the first part of this exercise, plot the points A(1, 2) and B(1, -3) on a graph. Both points have the same x coordinate and different y coordinates.

Step 2: Calculating the Slope

The slope of a line is given by the formula (y2 - y1)/(x2 - x1) where (x1, y1) and (x2, y2) are coordinates of two points on the line. In this case, however, as x1 equals x2, the formula will lead to a division by zero, which is undefined. Therefore, the line passing through these points is a vertical line and its slope will be undefined.

Step 3: Drawing the Graph of the Line

Based on the identified coordinates, draw a straight vertical line passing through the points A and B. This line represents the line L in the xy plane.

Key Concepts

Slope of a Line

To understand the slope of a line, imagine a road going uphill or downhill. The slope tells us how steep that road is. Mathematically, the slope of a line between two points, (x_1, y_1) and (x_2, y_2), can be calculated with the formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] This formula represents the change in y (vertical change) over the change in x (horizontal change).

• A positive slope means the line goes uphill.
• A negative slope means the line goes downhill.
• A zero slope indicates a perfectly horizontal line.

Visualizing this formula helps understand why the slope is a crucial concept in graphing. It informs us about the line's direction and steepness, making it easier to predict and understand the behavior of the line on a graph.

Undefined Slope

Sometimes, when we try to calculate the slope, we run into a situation where the formula doesn't work quite right. If the x coordinates of the two points are the same, this will cause a division by zero: \[ (x_2 - x_1) = 0 \] Since dividing by zero is undefined in mathematics, the slope here also becomes undefined. This scenario gives us a special type of line known as a **vertical line**. Because vertical lines do not rise or fall horizontally, they have no defined slope. Essentially, they go straight up or down. Vertical lines are fascinating in geometry because they break the usual rules of slopes by not really having one! It's important to remember that whenever points share the same x coordinate, the line through them will always be vertical.

Graphing Points

Graphing points is one of the basic tasks when dealing with the Cartesian plane. To graph a point, you need two pieces of information: an x and a y coordinate. These coordinates tell you the exact location of the point on the plane. Here's how to plot the points:

• Start at the origin, which is the point (0, 0) on the graph.
• Move horizontally to the x coordinate of your point. Positive values move right; negative values move left.
• Then continue vertically to the y coordinate. Positive values move up; negative values move down.
• Mark this spot on the graph, and you'll find your point's location.

For example, with points A(1, 2) and B(1, -3), you would first move to x = 1 and then move up to y = 2 for point A and down to y = -3 for point B.