<< Chapter < Page Chapter >> Page >
This graph shows the line y = 2 on an x.y coordinate plane.  The x-axis runs from negative 5 to 5 and the y-axis runs from – 5 to 5. A horizontal line crosses through the point (0, 2).  Underneath the graph is a table with two rows and six columns.  The top row is labeled: “x” and has the values negative 4, negative 2, 0, 2, and 4. The bottom row is labeled “y” and has the values 2, 2, 2, 2, and 2.
A horizontal line representing the function f ( x ) = 2

A vertical line    indicates a constant input, or x -value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.

This is an image showing when a slope is undefined.  m = change of output divided by the change of input.  The change of output is labeled as: non-zero real number and the change of input is labeled 0.
Example of how a line has a vertical slope. 0 in the denominator of the slope.

A vertical line, such as the one in [link] , has an x -intercept, but no y- intercept unless it’s the line x = 0. This graph represents the line x = 2.

This graph shows a vertical line passing through the point (2, 0) on an x, y coordinate plane. The x-axis runs from negative 5 to 5 and the y-axis runs from negative 5 to 5.  Underneath the graph is a table with two rows and six columns.  The top row is labeled: “x” and has the values 2, 2, 2, 2, and 2. The bottom row is labeled: “y” and has the values negative 4, negative 2, 0, 2, and 4.
The vertical line, x = 2 , which does not represent a function

Horizontal and vertical lines

Lines can be horizontal or vertical.

A horizontal line    is a line defined by an equation in the form f ( x ) = b .

A vertical line    is a line defined by an equation in the form x = a .

Writing the equation of a horizontal line

Write the equation of the line graphed in [link] .

This graph shows the function y = negative 4 on  an x, y coordinate plane.  The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The horizontal line passes through the point, (0, -4).

For any x -value, the y -value is 4 , so the equation is y = 4.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Writing the equation of a vertical line

Write the equation of the line graphed in [link] .

This is a graph showing a line with an undefined slope on an x, y coordinate plane. The x-axis runs from negative 10 to 10 and the y-axis runs from -10 to 10. The line passes through the point (7, 0).

The constant x -value is 7 , so the equation is x = 7.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Determining whether lines are parallel or perpendicular

The two lines in [link] are parallel lines    : they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y -intercept. If we shifted one line vertically toward the other, they would become coincident.

This graph shows two lines on an x, y coordinate plane. The x-axis runs from negative 4 to 6. The y-axis runs from negative 3 to 8.  The first line has the equation y = -3 times x divided by 2 plus 1.  The second line has the equation y = -3 times x divided by 2 plus 7.  The lines do not cross.
Parallel lines

We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y -intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.

f ( x ) = 2 x + 6 f ( x ) = 2 x 4 }  parallel f ( x ) = 3 x + 2 f ( x ) = 2 x + 2 }  not parallel

Unlike parallel lines, perpendicular lines    do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in [link] are perpendicular.

This graph shows two functions perpendicular to each other on an x, y coordinate plane. The first function increases and passes through the points (1, 0) and (0, -5).  The second function decreases and passes through the points (1, 0) and (-4, 1).  The lines intersect to form a 90-degree right angle at the point (1, 0).
Perpendicular lines

Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is 1. So, if m 1  and  m 2 are negative reciprocals of one another, they can be multiplied together to yield –1.

m 1 m 2 = −1

To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is 1 8 , and the reciprocal of 1 8 is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.

As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.

f ( x ) = 1 4 x + 2 negative reciprocal of 1 4  is  −4 f ( x ) = −4 x + 3 negative reciprocal of −4  is  1 4

The product of the slopes is –1.

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask