Horizontal and vertical lines are two special cases of linear equations in coordinate geometry. Let us learn about them in detail.
Horizontal Lines
A horizontal line is a straight line that runs from left to right, parallel to the x-axis in a coordinate plane. This means that all points on a horizontal line have the same y-coordinate.
Equation
The general equation of a horizontal line is:
y = b
Here,
b is a constant, representing the y-coordinate of every point on the line
It does not intersect the x-axis unless y = 0
Slope
Since all points on a horizontal line have the same y-coordinate, the slope equation is of the form:
y = c, here, c is a constant
Any two points on the line can be written as (x1, y) and (x2, y) and so on.
To find the slope m, we use the formula:
m = ${\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}}$
Substituting the values, we get
⇒ m = ${\dfrac{y-y}{x_{2}-x_{1}}}$
⇒ m = 0
Thus, the slope of a horizontal line is 0
Graphing
As we know, a horizontal line has an equation of the form y = c. This means that the y-coordinate stays the same for all points on the line, while the x-coordinate can be any real number.
To graph a horizontal line:
First, we identify the y-value from the given equation and locate this value on the y-axis
Then, we plot a straight line passing through the y-value and parallel to the x-axis.
The x-values can vary, but the y-coordinate remains the same for all points on the line.
For example, let us graph the equation y = 3
Here, every point on the line has a y-coordinate of 3, such as (-2, 3), (0, 3), (4, 3), and so on. Graphing this equation results in a straight line parallel to the x-axis at y = 3.
Applications
Symmetry: A horizontal line acts as a reflection axis across the x-axis, keeping the y-coordinate constant while allowing the x-coordinate to change. This property is useful in symmetry-related geometry problems.
Horizontal Line Test: A horizontal line is used to determine if a function is one-to-one. If it crosses a graph at multiple points, the function does not have a unique inverse.
Vertical Lines
A vertical line is a straight line that runs up and down parallel to the y-axis in a coordinate plane. All points on a vertical line have the same x-coordinate.
Equation
The general equation of a vertical line is:
x = a
Here,
a is a constant, representing the x-coordinate of every point on the line
Slope
Since all points on a vertical line have the same x-coordinate, the slope equation is of the form:
x = c, here c is a constant
Any two points on the line can be represented as (x, y1) and (x, y2).
To find the slope m, we use the slope formula:
m = ${\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}}$
Substituting the values, we get
⇒ m = ${\dfrac{y_{2}-y_{1}}{x-x}}$
Since division by zero is undefined, the slope of a vertical line is undefined.
Graphing
As we know, a vertical line has an equation of the form x = c. Thus, the x-coordinate remains the same for all points on the line, while the y-coordinate can take any value.
To graph a vertical line:
First, we identify the x-value from the given equation.
Then, we plot a straight line passing through that x-value and parallel to the y-axis.
Now, let us graph the equation x = -3
Here, every point on the line has an x-coordinate of -3, such as (-3, 1), (-3, 2), (-3, 3), and so on. Graphing this equation results in a straight line parallel to the y-axis at x = -3.
Applications
Symmetry: A vertical line serves as a reflection axis along the y-axis, keeping the x-coordinate fixed while changing the y-coordinate. This property is often seen in symmetrical shapes and graphs.
Vertical Line Test: This test helps determine whether a graph represents a function. If a vertical line cuts a graph at more than one point, the graph fails the test and is not a function, as a function must have only one output (y-value) for each input (x-value).
Summary
Basis
Horizontal
Vertical
Orientation
Left to right
Up and down
Equation
y = constant
x = constant
Slope
0
Undefined
Parallel to
x-axis
y-axis
Solved Examples
Write the equation of a horizontal line passing through (2, -4).
Solution:
As we know, a horizontal line has a constant y-coordinate for all points on the line. Since the given point is (2, -4), the y-coordinate is -4, the equation of the horizontal line is: y = -4 Thus, the equation of the horizontal line is y = -4
Write the equation of a vertical line passing through (-5, 7).
Solution:
As we know, a vertical line has a constant x-coordinate for all points on the line. Since the given point is (-5, 7), the x-coordinate is -5, the equation of the vertical line is: x = -5 Thus, the equation of the vertical line is x = -5
Determine whether the given equations represent a horizontal or vertical line: a) y = 10 b) x = -8
Solution:
a) Given, y = 10 This equation represents a horizontal line because the y-coordinate is constant for all points. Thus, it is a horizontal line. b) Given, x = -8 This equation represents a vertical line because the x-coordinate is constant for all points. Thus, it is a vertical line.
Find two points on the horizontal line y = -2.
Solution:
Since it is a horizontal line, the y-coordinate remains -2 for all values of x. Choosing two different x-values, we get: (3, -2) (-1, -2) Thus, two points on the line are (3, -2) and (-1, -2).
