## Slope Concept 3 – Slopes of Vertical and Horizontal Lines

Note: This is the third part of the Slope Concept Series.

Part I: Basic Understanding of Slope

In the Understanding the Basic Concepts of Slope post, we have discussed that the slope of a line in the coordinate plane is described as the change in y over the change in x. When we say change in x, or change in y, we talk about the ‘change distance.’ To determine a distance we need two points.  If we are in the coordinate plane, and we have two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$, then the rise is $y_2 - y_1$ and the run is $x_2 - x_1$. Thus, the slope of the line containing the two points is $\displaystyle\frac{y_2-y_1}{x_2-x_1}$.

Fiigure 1 - Slope of a line containing points with coordinates (x1,y1) and (x2,y2).

We also have discussed that the slope of a horizontal line is 0 by rotation. Here, we will show the same fact using coordinates as shown.

A horizontal line has the same y-coordinates everywhere. Let us consider line l in Figure 2 with equation $y = a$.   Let us pick two points with coordinates $(x_1,a)$ and $(x_2,a)$. Using the formula above, calculating for the slope we have $\displaystyle\frac{a -a}{x_2 - x_1} = \frac{0}{x_2 - x_1 }= 0$. Hence, the slope of a horizontal line l is $0$.

Figure 2 - A horizontal line containing points with coordinates (x_1,a) and (x_2,a)..

Similarly, a vertical line has the same x-coordinate everywhere. Let us consider line m in Figure 3 with equation $x = b$.   Let us pick two points with coordinates $(b,y_1)$ and $(b,y_2)$. Using the formula above, calculating the slope, we have $\displaystyle\frac{y_2 - y_1}{b - b} = \frac{ y_2 - y_1}{0}$ which is undefined.

Figure 3 - A horizontal line containing points with coordinates (b,y2) and (b,y1).

In the sequel of this post, we will discuss the lines with positive and negative slopes.

## Slope Concept 2 – Slope of the Graph of a Linear Function

Note: This is the second part  of the the Slope Concept Series. The second and third articles are Part I – Understanding the Basic Concepts of Slope and Part III – Slopes of Vertical and Horizontal Lines.

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In the Understanding the Basic Concepts of Slope post, we have discussed that slope is described as rise over run. In this post, we are going to show that the slope of a straight line is constant.

To get the slope of a straight line or a segment, we determine two points on the line, say A and B, draw a horizontal line through point A and a vertical segment through point B. We then determine the intersection of the two line segments and name it C as shown in Figure 1. Angle C is a right angle since BC is a vertical segment and AC is a horizontal segment.

Figure 1 - A line containing points A and B.

The slope of the line containing AB is $\frac{BC}{AC}$. If we determine two more points, say D and E on the line, and do the process mentioned above, we can come up with triangle DEF right angled at F as shown in Figure 2. In terms of DEF, the slope of the line containing points D and E is $\frac{EF}{DF}$. Since the line containing DE and AB is practically the same line, we have learned from high school mathematics that their slopes must be equal.  Hence, the following relationship holds: $\frac{BC}{AC} = \frac{EF}{DF}$.

Why is this so?

Figure 2 - Triangle ABC and DEF with their hypotenuse contained on the line.

To show that the slope of the line is constant, we must show that $\frac{BC}{AC} =\frac{EF}{DF}$.

Proof That the Slope of a Straight Line is Constant

From Figure 2, BC is parallel to EF, since they are both vertical segments.  Similarly, DF is parallel to AC, since they are both horizontal segments. By the Parallel Postulate, we can consider AB as a transversal of the two pairs of parallel segments.

We can see that angles DEF and ABC are congruent since they are corresponding angles. Angles C and F are also congruent since they are both right angles. Hence, by AA similarity, ABC is similar to DEF.  Since the ratio of the corresponding sides of a similar triangles are equal, it follows that $\frac{BC}{AC} =\frac{EF}{DF}$.

From the above proof, we have shown that the slope of a straight line, or the slope of the graph of a linear function,  is constant.

## Slope Concept 1 – Understanding the Basic Concepts of Slope

Note: This is the first part of the the Slope Concept Series. The sequels of this article are Part II – Slope of the Graph of a Linear Function and Part III – Slopes of Vertical and Horizontal Lines.

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The slope is known to be the steepness of a line.  Sometimes it is described as “rise over run,” If we are on point A, we go up 4 units and we go right 5 units (see Figure 1) then our rise is 4 and our run is 5. Let us mark our new location B. Notice that the order of movements does not matter. We can also go 5 units right and 4 units up and you will still be in B (see Figure 2).

If we do our movement in the coordinate plane starting from the origin, our rise would be our vertical movement (change of movement with respect to the y-axis) and our run would be our horizontal movement (change of movement with respect to the x-axis). In Figure 2, segment AB has rise 4 and run 5.  Thus, the slope of segment AB is $\displaystyle\frac{4}{5}$.  In general, slope in the coordinate plane is described as the change in y over the change in x.

Figure 1 - Segment AB with rise 4 units and run 5 units.

The slope of a line (or a segment) may also be described as the angle it makes with a horizontal line.  Technically speaking, it is a counterclockwise rotation with the line starting from a horizontal position about a point which is located on that line, or the origin our case.  In Figure 2, $\theta$ is the angle measure AB makes with the horizontal axis of the rectangular coordinate plane, or the amount of rotation from AB’ to AB about A.

Figure 2 - Counter-clockwise rotation of AB to AB' about A.

Looking at triangle ABC, since the given sides are the side adjacent and the side opposite to $\theta$, we can use the definition of tangent to compute for the value of $\theta$. Recall from trigonometry that the definition of a tangent of an angle of a right triangle is equal to the quotient of the length of the side opposite to it (change in y) and the length of the side adjacent to it (change in x). Now, this is precisely the definition of slope. From here, we can conclude that the angle that a line makes with a horizontal line is the same as the slope of that line. As a consequence $\tan(\theta) = \displaystyle\frac{4}{5}$ in radian measure (or approximately 38 degrees) is the slope of the line.

Figure 3 - Triangle ABC with Slope 4/5.

If we examine the value of $\theta$, it is clear that when $\tan(\theta)$ is $0$ degrees, the line is horizontal since there is no (zero) change in y. Algebraically, this makes the numerator of the fraction change in y $0$ which implies that the slope of any horizontal line is $0$.

If the line is vertical, there is no (zero) change in x. That makes the denominator of the fraction change in x $0$. Of course, we know that anything divided by $0$ is not defined. As a consequence, slope of a vertical line is undefined.

In the continuation of this article, we will discuss further about the properties of slope. We will discuss why the slope of a straight line is constant. We will further discuss zero, undefined, negative and positive slopes. We will also discuss how the concept of slope helps in solving calculus problems and how it is used to determine the behavior functions.