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8

THE SLOPE
OF A STRAIGHT LINE

Definition of the slope


The slope of a straight line.

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

Change in y-coördinate
Change in x-coördinate
 =   62
31
  =   4
2
  =  2

Here is another example,

Slope

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

  that line is  2
3
.

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

Up or down?

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.

Positive slope. Negative slope.

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.

Up or down?

To see the answers, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

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?

Δy
Δx
 =   −8 − (−2)
  7 − (−1)
  =   −8 + 2
  7 + 1
  =   −6
  8
  =  −  3
4

This means that for every 4 units the line goes over, it goes down 3.

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, Δyx = (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.

f(x + h) − f(x)
         h

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

To establish the algebraic equation of any geometrical figure, we must carry over our knowledge from plane geometry; for nothing will come of nothing.  With the aid of the following theorems, we will be able to prove that the equation of a straight line may take the form  y = ax + b,  and that a is the slope of the line.  We will do that in the following Topic.

First,


Theorem 8.1.   A straight line has one slope.

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;

therefore, the exterior angle DCF is equal to the opposite interior angle on the same side, angle BAE.  (Euclid, I. 29)

Therefore the right triangles BAE, DCF are similar (Euclid, VI. 4), and therefore, proportionally,

BE
EA
  =   DF
FC
 .
Δy
Δx
will be the same, whether we go from A to B, or from C to D.

This is what we wanted to prove.


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.

Same slope means Parallel

Let the straight lines L1, L2 be parallel.  And let the slope of L1 be BC/CA;  for a straight line has only one slope.  (Theorem 1)

Draw the straight line ACF intersecting L2 at D, and draw FE perpendicular to DF.

Then the slope of the straight line L2 is EF/FD.  

We claim that the slope of L2 is equal to the slope of L1.  That is,

BC
CA
  =   EF
FD

For, since the lines L1, L2 are parallel, and the straight line AF falls on them, the alternate angles BAC, EDF are equal  (Euclid, I. 29);

therefore the right triangles BAC, EDF are similar  (Euclid, VI.4),

and those sides are proportional that contain the equal angles:

BC
CA
  =   EF
FD

Therefore the straight lines L1, L2 have the same slope.

Next, assume that those lines have the same slope:

BC
CA
  =   EF
FD

Same slope means Parallel

Then the two straight lines are parallel.

For, the angles at C and F are right angles and therefore equal,

and by assumption, the sides that make them are proportional.

But if two triangles have one angle equal to one angle, and the sides that make them are proportional, then the triangles are similar;
and those angles are equal that are opposite the corresponding sides;
(Euclid, VI. 6, 4)

therefore, angle BAC is equal to angle EDF.

But angle CDG is equal to angle EDF, because they are vertical angles;  (Euclid, I. 15)

therefore angle BAC is equal to angle CDG.

That is, the straight line AF falling on the straight lines L1, L2 makes the alternate angles BAC, CDG equal.

Therefore the straight line L1is parallel to the straight line L2.
(Euclid, I. 27)

Therefore two straight lines are parallel if and only if they have the same slope.  Which is what we wanted to prove.

As for perpendicular lines, in Lesson 34 of Algebra we prove the following:

If two straight lines are perpendicular to one another, then the product of their slopes is −1.

That is, if the slope of one line is m, then the slope of the perpendicular line

   is −  1 
m
.

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.

Therefore, any two lines through P will not be parallel, and therefore they will have different slopes.  (Theorem 2)

Therefore, there is only one straight line through P with a given slope.

This means that a straight line may be specified by giving its slope and the coördinates of one point on it.

Which is what we wanted to prove.

*

On the basis of this theorem, we will be able to  prove, in the following Topic, that the equation of a line may take the form y = ax + b, where a is the slope of the line, and b is the y-intercept.


Next Topic:  Linear functions


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