Horizontal line

Horizontal line

Do you know what is meant by horizon?

Horizon is a line along which the earth's surface and the sky appear to meet (but they do not actually meet).

Horiontal line introduction: Horizon image

The word "horizontal" is derived from the word "horizon".

"Horizontal" means "side-to-side".

Horizontal lines are lines that are parallel to the horizon.

Let's learn in detail about horizontal lines.

Lesson Plan 

What Are Horizontal Lines?

Horizontal Line Definition

  • In general, horizontal lines are sleeping lines.
  • Horizontal lines are lines that are parallel to the horizon.
  • Horizontal lines in coordinate geometry are lines that are parallel to the x-axis.

What are horizontal lines? Horizontal lines are sleeping lines.


Horizontal Line Images

Horizontal Line Examples in Real Life

Here are some horizontal line examples in real life.

Other popular examples include steps on the staircase, planks on the railway tracks, etc.

Horizontal line images: A man lying horizontally

Horizontal lines of alphabets A, L, and T

Horizontal Line Examples in Geometry

In geometry, we can find horizontal line segments in many shapes, such as quadrilaterals, 3d shapes, etc.

In coordinate geometry, horizontal lines are lines that are parallel to the x-axis.

Here are some horizontal lines on a coordinate plane.

Horizontal lines in coordinate geometry


What Are Horizontal and Vertical Lines?

  • "Horizontal" means "side-to-side" and a horizontal line is nothing but a sleeping line.
  • "Vertical" means "up-to-down" and a vertical line is nothing but a standing line.

"Horizontal" and "vertical" are opposite words of each other.

Horizontal lines and vertical lines are perpendicular to each other.

Horizontal and vertical lines


How Do You Make a Horizontal Line?

To make a horizontal line on plain paper, you can use a ruler. Place it parallel to the horizontal edge of the paper and draw a line along the ruler.

But how to draw a horizontal line on a coordinate plane? Let's see.

To draw a horizontal line, 

  • Place a dot at any random point on the coordinate plane, let's say at (2, -3).
  • Identify its y-coordinate. Here the y-coordinate is -3.
  • Plot some other points whose y-coordinate is the same as the point plotted. Let's plot (1, -3), (-2, -3), etc.
  • Join all the points and extend it on both sides to get the horizontal line.

How to make a horizontal line on a coordinate plane?


Horizontal Line Equation

In the last section, we can see that the y-coordinates of all the points on a horizontal line are the same.

Thus, the equation of a horizontal line through any point \((a,b)\) is of the form:

  Horizontal line equation

Here \(x\) is absent. It means that the x-coordinate can be anything whereas the y-coordinate of all the points on the line must be \(b\) only.

The horizontal line slope is 0 as by comparing \(y=b\) with \(y=mx+b\), we get the slope to be \(m=0\).

Here is an example:

Graph of a constant function f(x) = 3

Take a look at some of the points on this line: (-5, 3), (-1, 3), (4, 3), (7, 3).

You can see that the y-coordinate of all these points is constant, which is 3. Hence the equation of this line is \(y=3\).


How to Use the Horizontal Line?

Horizontal Line of Symmetry

Horizontal lines are used to represent symmetry.

Horizontal line of symmetry is a horizontal line that makes a shape exactly into two equal parts such that when the shape is folded along that line, one part overlaps the other.

In each of the following figures, the dotted line is a horizontal line of symmetry.

Horizontal line of symmetry

Horizontal line of symmetry

Horizontal Line Test

The horizontal line test is used to determine whether a function is one-one.

According to the horizontal line test, a function is NOT one-one if there exists a horizontal line that passes through more than one point of the graph (of the function).

Example 1

Horizontal line test

Here, \(f(x)\) is one-one because every horizontal line passes through at most one point of the graph.

Example 2

Horizontal line test

Here, \(g(x)\) is not one-one as there exists a horizontal line that passes through more than one point of the graph.

 
important notes to remember
Important Notes
  1. A horizontal line (except x-axis) has no x-intercepts.
  2. The equation of a horizontal line passing through \((a, b)\) is \(y = b\).
  3. The equation of a vertical line passing through \((a, b)\) is \(x = a\).
  4. The slope of a horizontal line is 0 as its equation is of the form \(y=b\) and comparing it with the slope-intercept form, we get its slope to be \(m=0\).
  5. The vertical line test is used to determine whether a relation is a function whereas the horizontal line test is used to determine whether a function is one-one.

Solved Examples

Example 1

 

 

Can we help Jake find the equation of the following line?

Example of the horizontal line: A graph showing a vertical line

Solution

The given line is a horizontal line passing through the point: \[(a,b)=(1,3)\]

We know that the equation of a horizontal line passing through a point \((a,b)\) is \(y=b\).

Therefore, the equation of the given line is:

\(\therefore\) \(y=3\)
Example 2

 

 

Can we help Ava identify the capital alphabets that have only a horizontal line of symmetry but no vertical line of symmetry?

Solution

Let us identify the symmetry (horizontal/vertical) of each alphabet.

Alphabets with lines of symmetry

We can see that only the alphabets B, C, D, E, and K have only a horizontal line of symmetry but no vertical line of symmetry.

Therefore, the required alphabets are:

\(\therefore\) B, C, D, E, and K
Example 3

 

 

Can we help Mia find which of the following functions have an inverse?

a) \(f(x) = x^3\)

b) \(g(x) = x^2\)

c) \(h(x) = |x|\)

d) \(k(x) = \ln x\)

Solution

Let us graph each of these functions and see which of them would pass the horizontal line test.

We draw a horizontal line to see how many graphs have at most one point of intersection with the horizontal line.

Horizontal line example: Which of the following functions has inverse?

Here, each of \(h\) and \(g\) has two points of intersection with the horizontal line and hence they are not one-one. Thus they can't have an inverse.

Whereas each of \(f\) and \(k\) has only one point of intersection with the horizontal line.

Hence, \(f\) and \(k\) are one-one and hence each of them has an inverse. Therefore,

\(\therefore\) Only \(f\) and \(k\) have inverse

Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
tips and tricks
Tips and Tricks
  1. Whenever we see an equation of the form \(y=\) a constant, we must realize that it represents the equation of a horizontal line.
  2. An equation of the form \(y=\) a constant represents a constant function.
  3. To determine whether a function has an inverse, use the horizontal line test (because for a function to have an inverse, it has to be one-one).

Let's Summarize

The mini-lesson targeted the fascinating concept of Horizontal Line. The math journey around the Horizontal Line starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Horizontal Line

1. Does a horizontal line have a slope?

No, a horizontal line has no slope, i.e., the slope of a horizontal line is 0.

2. What is the horizontal and vertical line?

  • A horizontal line is a line parallel to the x-axis and its equation is of the form \(y=\) a constant.
  • A vertical line is a line parallel to the y-axis and its equation is of the form \(x=\) a constant.

3. Which is a vertical line?

A vertical line is a line parallel to the y-axis and its equation is of the form \(x=\) a constant.

4. How is a horizontal line drawn?

To draw a horizontal line on paper, use a ruler. Place it parallel to the horizontal edge of the paper and draw a line along the ruler.

To draw a horizontal line on a coordinate plane,

  • Place a dot at any random point on the coordinate plane, let's say at (2, -3).
  • Identify its y-coordinate. Here the y-coordinate is -3.
  • Plot some other points whose y-coordinate is the same as the point plotted. Let's plot (1, -3), (-2, -3), etc.
  • Join all the points to get the horizontal line.

5. Which is an equation of a horizontal line?

The equation of a horizontal line passing through a point (a, b) is y = b.

6. Is a horizontal side to side or up and down?

As we learned on this page, a horizontal is a side-to-side line.

7. What is meant by a horizontal line?

  • Horizontal lines are lines that are parallel to the horizon.
  • Horizontal lines in coordinate geometry are lines that are parallel to the x-axis.
  • Horizontal lines are sleeping lines.

8. What is an example of vertical?

An example of "vertical" is an electric pole that is perpendicular to the ground.

9. What are the horizontal lines on the globe called?

The horizontal lines on the globe are called "latitudes."

10. What are the vertical lines on the globe called?

The vertical lines on the globe are called "longitudes."

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