1. Home
  2. Algebra and trigonometry
  3. Linear functions
  4. Linear functions

4.1 Linear functions  (Page 12/27)

<< Chapter < Page
  Algebra and trigonometry     Page 12 / 27
Chapter >> Page >
− 4 ( 1 4 ) = − 1

Parallel and perpendicular lines

Two lines are parallel lines    if they do not intersect. The slopes of the lines are the same.

f ( x ) = m 1 x + b 1   and   g ( x ) = m 2 x + b 2   are parallel if and only if  m 1 = m 2

If and only if   b 1 = b 2   and   m 1 = m 2 ,   we say the lines coincide. Coincident lines are the same line.

Two lines are perpendicular lines    if they intersect to form a right angle.

f ( x ) = m 1 x + b 1   and   g ( x ) = m 2 x + b 2   are perpendicular if and only if
m 1 m 2 = − 1 , so   m 2 = − 1 m 1

Identifying parallel and perpendicular lines

Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

f ( x ) = 2 x + 3 h ( x ) = − 2 x + 2 g ( x ) = 1 2 x − 4 j ( x ) = 2 x − 6

Parallel lines have the same slope. Because the functions   f ( x ) = 2 x + 3   and   j ( x ) = 2 x − 6   each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and   1 2   are negative reciprocals, the functions   g ( x ) = 1 2 x − 4   and   h ( x ) = −2 x + 2   represent perpendicular lines.

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

Writing the equation of a line parallel or perpendicular to a given line

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

Writing equations of parallel lines

Suppose for example, we are given the equation shown.

f ( x ) = 3 x + 1

We know that the slope of the line formed by the function is 3. We also know that the y- intercept is   ( 0 , 1 ) .   Any other line with a slope of 3 will be parallel to   f ( x ) .   So the lines formed by all of the following functions will be parallel to   f ( x ) .

g ( x ) = 3 x + 6 h ( x ) = 3 x + 1 p ( x ) = 3 x + 2 3

Suppose then we want to write the equation of a line that is parallel to   f   and passes through the point   ( 1 , 7 ) .   This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of   b   will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.

y − y 1 = m ( x − x 1 ) y − 7 = 3 ( x − 1 ) y − 7 = 3 x − 3 y = 3 x + 4

So   g ( x ) = 3 x + 4   is parallel to   f ( x ) = 3 x + 1   and passes through the point   ( 1 , 7 ) .

Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point.

  1. Find the slope of the function.
  2. Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.
  3. Simplify.

Finding a line parallel to a given line

Find a line parallel to the graph of   f ( x ) = 3 x + 6   that passes through the point   ( 3 , 0 ) .

The slope of the given line is 3. If we choose the slope-intercept form, we can substitute   m = 3 , x = 3 , and   f ( x ) = 0   into the slope-intercept form to find the y- intercept.

g ( x ) = 3 x + b 0 = 3 ( 3 ) + b b = –9

The line parallel to   f ( x )   that passes through   ( 3 , 0 )   is   g ( x ) = 3 x − 9.

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

Writing equations of perpendicular lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 10

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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