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.
We will start with some simple examples
and focus attention on the slopes of these lines.
A line with positive slope
A line with zero slope
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=(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