CONSIDER THIS STRAIGHT LINE, and let us move from a point A on the line to a point B.  Then the coördinates will change.  By the symbol Δx ("delta x") we mean by how much the x-coördinate will change (as we move from A to B).  And by the symbol Δy ("delta y") we mean by how much the y-coördinate will change.

Then by the slope of the line we mean the number

Δy Δx

=

=

Change in y-coördinate Change in x-coördinate

For example, if the coördinates at B are (3, 6),

and the coördinates at A are (1, 2),

then the slope of that line is

Here is another example,

If, to get from A to B, we move over 3 and up 2, then the slope of

That means that for every 3 units the line moves over, it moves up 2.

Up or down?

Which line is sloping "up"?  And which line is sloping "down"?

The line on the left is sloping "up," and its slope will be a positive number.  The line on the right is sloping "down."  Its slope will be a negative number.

For, as we move from A to B, the x-coördinate at B will be greater than the x-coördinate at A:  Δx will be positive.   And the y-coördinate at B will also be greater than at A:  Δy will also be positive.

But as we move from C to D, although Δx will be positive, the y-coördinate at D will be less than the y-coördinate at C.  Δy will be negative.

Specifically, if to get from A to B, we go over 1 and then up 1, then Δx = 1, and Δy = 1, so that the slope of that line is 1/1 = 1.

But to get from C to D, we must go over 1 and then down 1.   Δx = 1, but Δy = −1 -- because the y-coördinate at D will be less than the y-coördinate at C.  The change in that y-coördinate will be negative.  Therefore, the slope of that line is 1/−1 = −1.

The point is:

A line that is sloping down has a negative slope.

Horizontal and vertical lines

What is the slope of a horizontal line -- that is, a line parallel to the x-axis?  And what is the slope of a vertical line?

A horizontal line has slope 0, because as we move along the line, the y-coördinate does not change.  Δy = 0.

For a vertical line, however, the slope is not defined.  The slope tells how the y-coördinate changes when the x-coördinate changes.  But the x-coördinate does not change -- Δx = 0.  A vertical line does not have a slope.

(In a vertical line, the denominator of the slope, Δx, would be 0.  And a symbol with denominator 0 does not signify a number.)

Problem 1.

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a)  Which numbered lines have a positive slope?   2 and 4.

b)  Which numbered lines have a negative slope?   1 and 3.

c)  What slope has the horizontal line 5?   0.

c)  What slope has the vertical line 6?   It does not have a slope.

Problem 2.   Calculate the slope of the straight line joining the points (−1, −2) and (7, −8).  And what does that slope mean?

Problem 3.   Let y = f(x) = x² − 1, and let two points on its graph have these x-coördinates:  x₁ = 3,  x₂ = 5.  Calculate the slope of the straight line that passes through those two points.

At each of those points, f(x) will give the value of y. Thus, at x₁ = 3,  y₁ = 3² − 1 = 8.  At x₂ = 5,  y₂ = 5² − 1 = 24.  Therefore, Δy/Δx = (24 − 8)/(5 − 3) = 16/2 = 8.

Problem 4.

Let y = f(x), and let its graph pass through two points A and B whose x-coördinates are separated by a distance h.  And let x be the x-coördinate of A.

a)  What is the x-coördinate of B?   x + h

b)  What are the x- and y-coördinates of A?   (x, f(x))

c)  What are the coördinates of B?   (x + h, f(x + h))

d)  Write an expression for the slope of the straight line joining the
d)  points A and B.

This is the famous "difference quotient." It is the slope of the line joining two points on a curve. In calculus, this will be fundamental.

One slope

Theorem 8.1.   A straight line has one slope.

That is, whether we measure the slope by moving from point A to point B, or by moving from point C to point D, the number we obtain will be the same.

For, if we measure the slope from A to B, we will get BE/EA.  While if we measure it between C and D, we will get DF/FC.

But CF is parallel to AE, because they are both parallel to the x-axis;

and AD is a straight line that falls on two parallel straight lines;

Problem 5.   A straght line will make an angle θ with the x-axis.

Which trigonometric function of θ is its slope?

tan θ. For, a line has one slope. Therefore if (x, y) is any point on the line, then its slope is y/x = tan θ.

"Same slope" and "parallel"

Before we state the next theorem, the student should be familiar with the logical expression if and only if.

First, if p and q are statements (sentences), then in the proposition "If p, then q,"  p is called the hypothesis.  It is what we are given, or what we assume.  There is no logic without hypotheses.

Statement q is called the conclusion.  It is what we must prove on the basis of the hypothesis.

The proposition  "If q, then p"  is called the converse of  "If p, then q."   The hypothesis and conclusion are exchanged.

Finally, the proposition  "p if and only if q"  means: "If p, then q , and if q, then p."  In other words, a proposition and its converse are both true.

For example, "A triangles is isosceles if and only if the base angles are equal."  This means, "If a triangle is isosceles, then the bases angles are equal; and, conversely, if the base angles are equal, then the triangle is isosceles."

Therefore, when we prove an if and only if proposition, we must prove both a proposition and its converse.

Now, the slope of a line is a number.  Parallel is a geometrical property.  The relationship between them is the next theorem.

Theorem 8.2.   Two straight lines are parallel if and only if they have the same slope.

The specification of a straight line

Finally, we can state

Theorem 8.3.   A straight line may be specified by giving its slope and the coördinates of one point on it.  Equivalently:  Through any point there is exactly one straight line with a given slope.

Let the straight line L have slope m, and let it pass through the point P.  Then L is the only straight line through P with slope m.

For, it will be impossible to draw another straight line through P parallel to L, because parallel lines do not meet.