8 junio, 2024

Vertical line: characteristics and use in mathematics (examples)

We explain what a vertical is, its characteristics and applications in mathematics.

What is a vertical line?

A vertical line It is the one that follows the direction in which any object falls to the ground when it is released from a certain height and is perpendicular to the horizon line, since it forms an angle of 90º with it.

When drawing it, a stroke is made from top to bottom or vice versa. The side edges of a computer monitor screen are examples of vertical lines, as are the straight trunks of many trees.

In architecture and design, the vertical line suggests in people a sense of dynamism, movement, power and elevation, in contrast to the horizontal lines, which suggest rest and relaxation. When someone is upright, that is, their position is vertical and perpendicular to the ground, they are ready to walk, run and in general move.

You can find a multitude of vertical lines in art, photographs and human constructions, permanent or temporary lines, such as those formed by contrasts between light and shadow on the walls, throughout the day.

The vertical line is also used to describe a very common movement in nature: free fall, as well as to describe the direction of other forces, apart from the aforementioned gravity, when they act perpendicular to a given surface.

Mathematical form of the vertical line

In Mathematics and Geometry, the vertical line coincides with the Cartesian «y» axis, the axis of the dependent variable, while the horizontal axis corresponds to the «x» axis, that of the independent variable.

A vertical line can be easily graphed in the Cartesian plane, since it corresponds to the equation:

x=k

where k is a constant. Vertical lines are always parallel to the y-axis, for example the line x = −3 that appears in red in the following figure:

Note that all the points on this line always have the same x-coordinate, for example the points (−3, 0); (−3, 1), (−3, 2) and more. In addition, the line in red color intersects the horizontal axis at the coordinate x = −3.

On the other hand, the equation line x = 0 is another way of expressing the vertical axis or y axis.

slope of a vertical line

A vertical line is considered to have no defined slope, or it can also be said that the vertical line has infinite slope, while the slope of a horizontal line is 0.

When trying to use the formula to calculate the slope of a line: m = Δy/ Δx when calculating the slope of the vertical line, it happens that Δx is always equal to 0, since any point that is chosen has the same x-coordinate . Remember that Δx = x2 – x1, that is, the difference between the x coordinates of two arbitrary points.

So, trying to plug Δx = 0 into the slope equation, we find that:

m = Δy/ 0

And since division by 0 is not a definite operation, it turns out that the slope of any vertical line is undefined, regardless of the value of Δy.

Vertical line test

Unlike the horizontal line, which is the graph of the constant function, the vertical line x = k is not a function, since the same value of x forms infinitely many ordered pairs with the values ​​of y, which goes against the function definition (in this, a value of x has one y only one image in y).

However, the vertical line can be used to visually determine whether or not a curve in the plane is a function. The criterion is very simple: draw a vertical that cuts the curve in question. If it does at more than one point, it is not a function.

For example, consider the curve shown below, which you want to know if it corresponds to the graph of some function.

The same vertical line passes through the red points and since it intersects the curve in more than one point, it can be concluded that it is not the graph of a function.

vertical asymptotes

They are vertical lines that the graph of a function cannot traverse. They arise because when it approaches a certain value of x, the function increases or decreases indefinitely. Of course, this value of x does not belong to the domain of the function.

When dealing with a rational function, the values ​​of x that give rise to vertical asymptotes are those that cancel out the denominator. In this case, when trying to substitute that value of x in the function, a division by 0 would remain, which is not possible, as explained above.

Now, what it is possible to do is divide a finite quantity by another quantity as small as you want, as long as the quantity is not exactly 0.

In such cases, the result of the division can be an extremely large number (or small because it is negative, depending on the sign of the numerator). The reader can check this by dividing for example:

2 ÷ 0.000001 = 2,000,000

Suppose that the value of x that cancels out the denominator of the rational function is x = b. When a value very close to b (but not exactly b) is substituted into the function, it results in a division between a finite quantity and an extremely small quantity.

That is why the rational function tends to positive infinity or negative infinity near the vertical asymptote, depending on the value of x used to approach b.

Example of vertical asymptote

The above is verified with the rational function:

The value that cancels out the denominator is x = 2, therefore the function has a vertical asymptote on the line x = 2. Suppose you want to get closer to x = 2 by taking a slightly smaller value, for example x = 1.9999:

This was an approach to x = 2 from the left and the result is that the function becomes very negative, that is, it tends to negative infinity. Now it can be tried with a right-hand approach, for example x = 2.0001:

And we see that the function recedes towards positive infinity. The graph confirms it:

References

Atlantic Union Conference Teacher Bulletin. Horizontal and vertical lines. Retrieved from: teacherbulletin.org.
Byju’s. Vertical line. Recovered from: byjus.com.
CK-12. Graphing horizontal and vertical lines. Retrieved from: ck-12.org.
Stewart, J. 2006. Pre-calculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.
Zill, D. 1984. Algebra and Trigonometry. 1st Edition. McGraw Hill.

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