Horizontal Line

Horizontal Line Definition:

The word horizontal is derived from the word ‘horizon’, which means a line parallel to Horizon. Horizon is the line where the sky and the land seem to meet. Horizontal lines run straight from left to right. Example: Edges of staircases that looks like straight lines parallel to the horizon.

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In geometry, a horizontal line is a straight line that moves from left to right on a plane. In co-ordinate geometry, if two points on a line fall on the same y-coordinate, it will be a horizontal line. The horizontal line has a slope that evaluates “0”. The slope does not rise or fall or rise as we move to the right along the line. It is a parallel line to ground level.

Eg., The line separating the sky from the water at the beach.


Horizontal Line Equation:

Horizontal lines are always parallel to the x-axis and these lines intersect or cross with the y-axis. Their values never change. The “x” value varies but the “y” value remains the same. The slope of Horizontal lines is always zero.

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In the intercept equation of slope which is (mx+b), if we put m=0(as, for the horizontal line the value of slope is zero) the equation becomes y=b.

Here be is the y-intercept of y-coordinate.


b is a real number. All points that lie on the line have a “y” coordinate of b.

The X and Y term tells a specific coordinate point on the horizontal line. 


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Example 1: Write the following equation for the line that is passing through (6, 5). 

Y is constant, in the horizontal line, i.e., “Y” carries constant value on x-axis.

Since, “Y” takes the positive value of 5 at the point of intersection (6, 5)

So, the equation is Y = 5


Symmetry is a topic that comes up in mathematics and it is also very useful in everyday life. Especially, in the topics of art and design, the general idea behind symmetry is that, we can clarify that the shape has symmetry. If one half of the shape is the mirror image of the other half, it will also be called as symmetrical. 


Let’s take an example of a letter “B”. It looks like a horizontal symmetry and the top half is the mirror image of the bottom half. The symmetrical line produces equal reflections that coincide with each half. The slope in such a line is always “0”.


Examples:

Capital letters that have horizontal symmetry are:


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Horizontal Line Test:


Let’s use a linear function. For example, f(x) has an inverse function to determine. If f(x) has an inverse function we have to show it is a one–to–one function. We have to show that it passes the horizontal test. 

So, if we draw a horizontal line in which function of “x” touches the line more than once. i.e., f(x) is not a one-to-one function. But, if it intersects the horizontal line only at one point, implies that f(x) is a one-to-one function. This means, it has an inverse function.

 To find out if it actually has an inverse function, at that time we use a horizontal line test and see if for every “y” value, there is only one “x” value.


Illustration of Horizontal Line Segment


When a point of a segment divides that line segment into two congruent segments, the point is called a midpoint. 


Example (a):

A Horizontal Line that insects through the point (2, -1). Write the equation.

Y = b


Solution: 

(2, -1)  (2 = x co-ordinate, -1 = y co-ordinate)

 So, plug in -1 = b

Equation, y = -1

In Horizontal lines only have 'x' value and no 'y' value.


Example (b):

Determine if the line shown in Fig (a) is horizontal and write its equation.


The two points A, B on the line are at (7,7)

and (9,7). The second coordinate in each 

pair of y coordinate are 7 and 7.

Since they are equal, the line is horizontal.


Since the line passes the “y” axis at 7, the equation of the line,  y = 7

It can be read as "for all values of x, y is 7".


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Practice Questions

Q1. Which one is a horizontal line? 

Options:


(a)y=2x+1


(b)y=−x


(c) y=0 Correct answer: (c)  y=0


(d)y=3x+4


(e)x=4


Explanation:


A horizontal line has numerous values for x, but one possible value for y. Thus, it is always in the form y=c, Here “C” is a constant. The equation is y=0 (slope in horizontal line = 0)


Q2. Which of the following one is an equation of a horizontal line? 

Options:


(a)y=2x+4


(b)y=−2


(c)x=5       Correct answer: (b)y=−2


(d)y=x


(e)x=y2


Explanation:

In the horizontal line, the x value changes to be any real number, but the y value always remains constant. Thus, an equation in which the y value is constant and the x value is not present, this would be the equation in the form y=c, where “C” is a constant.


Q3. What is the equation for the horizontal line having the point (7, 10)?

Options:


(a)y=10


(b)x=7


(c)y=7 Correct answer: (d)y=10


(d)x=10


Explanation:

The equation for a horizontal line is y=b

Where, b is the y-coordinate of the point (a, b) on the line.

As such, the equation for the line contains the point (7, 10) is y=10