A Review of Lines and Slopes

This page serves as a quick review of straight lines and their important features. Many of these features are fundamental to a mathematical understanding of Calculus. After all, the branch of Calculus we will first explore (the part involving derivatives) is the study of how a curve can be approximated (at least in a very local sense) by a straight line (the tangent line). Thus, before we can discuss such concepts effectively, we must understand the idea of slopes and lines very, very well.

The reason we are interested in straight lines is that they are so simple; we need to know only two pieces of information to describe a line completely. Which two pieces of information are used depend on the situation, but several different possibilities lead to the same description of a particular line. We will explore these details below as we discuss

• The slope of a straight line
• How to describe a straight line

Because this is a review, most students can skim this material, making sure that they can answer the problems at the bottom of the page. Should you have difficulty in understanding or answering these questions, we suggest reviewing the material on equations of lines in your favourite Calculus or analytic geometry book.

What is slope?

Here is a simple demonstration showing a board leaning against a box. You can click on the bottom end of the board and drag it around. As you do so, notice that the steepness of the board varies. If you drag the end of the board all the way to the left, the board is not very steep. However, as you drag the end of the board to the right, it becomes much steeper. We would like some way to measure how steep the board is, and this is why slope enters our picture. Suppose the end of the board is 1m from the base of the box and we want to walk up the board. We need to walk up one metre for every metre we walk in the horizontal direction. The ratio of these two quantities determines the steepness of the board and so we define the slope to be the ratio of the vertical change to the horizontal change. In this case, 1m/1m = 1. Now when the end of the board is 3m from the base of the box, it is not nearly so steep. To walk up one metre, we need to walk 3m in the horizontal direction. Here the slope is 1m/3m = 1/3.

Positive and Negative Slopes We will start with some simple examples and focus attention on the slopes of these lines.

y=x A line with positive slope	y=1 A line with zero slope	y=-x A line with negative slope

We notice that the height of the line with positive slope increases if we move in the positive x direction (towards the right). (This innocuous statement will actually be very useful when we deal with derivatives a little bit later on.) Similarly, the line with negative slope has a decreasing height. The line y=1 has zero slope (there is no change in y as x changes), and thus no increase or decrease is occurring. Slopes thus give us information about the way that the graph of a function, in this case of a rather simple function, is changing. We will come back to this concept several times in this course. But first, let's be a little more precise and define what we actually mean by slope.

Definition of the slope of a straight line

Choose any two points on the graph of the line. Then the slope of the line, which we will call m is: m=(change in y value)/(change in x value) For your consideration: (1) Use this definition to determine slopes of the lines y=-x and y=1 (where C is a constant). (2) Determine the slope of the line shown in the graph. The fact that the slope can be defined is the defining feature of lines. In the next demonstration, you can see that, for a line, the ratio of the change in y to the change in x is always the same no matter which two points are chosen. However, with the parabola, the ratio clearly depends on which two points are chosen. To use the demonstration, click the mouse anywhere on the graph and then drag to see the ratio of the y change to x change. To change to the parabola, click on the "Parabola" button.