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

4.1 Linear functions  (Page 3/27)

<< Chapter < Page
  Algebra and trigonometry     Page 3 / 27
Chapter >> Page >
This figure shows three graphs labeled a, b, and c. Graph a shows an increasing function (f) along the x-axis and the y-axis which is labeled f of x. Graph b shows a decreasing function (f) along the x-axis and y-axis which is labeled f of x. Graph c shows a constant function (f) along the x-axis and y-axis which is labeled f of x. The constant function is horizontal. None of the graphs have any increments labeled on the x- or y-axis.

Increasing and decreasing functions

The slope determines if the function is an increasing linear function    , a decreasing linear function    , or a constant function.

  • f ( x ) = m x + b   is an increasing function if   m > 0.
  • f ( x ) = m x + b   is a decreasing function if   m < 0.
  • f ( x ) = m x + b   is a constant function if   m = 0.

Deciding whether a function is increasing, decreasing, or constant

Some recent studies suggest that a teenager sends an average of 60 texts per day http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/ . For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.

  1. The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.
  2. A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.
  3. A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.

Analyze each function.

  1. The function can be represented as   f ( x ) = 60 x   where   x   is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.
  2. The function can be represented as   f ( x ) = 500 − 60 x   where   x is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after   x   days.
  3. The cost function can be represented as   f ( x ) = 50   because the number of days does not affect the total cost. The slope is 0 so the function is constant.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Interpreting slope as a rate of change

In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,   x 1 and   x 2 , and two corresponding values for the output,   y 1   and   y 2   —which can be represented by a set of points,   ( x 1 ,  y 1 )   and   ( x 2 ,  y 2 ) —we can calculate the slope m .

m = change in output (rise) change in input (run) = Δ y Δ x = y 2 − y 1 x 2 − x 1

Note that in function notation we can obtain two corresponding values for the output   y 1   and   y 2   for the function   f ,   y 1 = f ( x 1 )   and   y 2 = f ( x 2 ) , so we could equivalently write

m = f ( x 2 ) – f ( x 1 ) x 2 – x 1

[link] indicates how the slope of the line between the points,   ( x 1 , y 1 )   and   ( x 2 , y 2 ) ,   is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.

This graph shows how to calculate the slope of a line. The line is graphed on an x y coordinate plane. The x-axis is labeled from negative 1 to 6. The y-axis is labeled from negative 1 to 10. The line passes through several points, but two are marked specifcally. The first is labeled (x subscript 1, y subscript 1). It is located at the point (1, 5). The second point is labeled (x subscript 2, y subscript 2). It is located at the point (2, 8). There is a small arrow that runs horizontally from point (2, 8) to point (1, 8). This arrow is labeled x subscript 2 minus x subscript 1. There is a blue arrow that runs vertically from point (1, 5) to point (1, 8) and is labeled y subscript 2 minus y subscript 1. Off to the side is the equation m equals delta y divided by delta x which equals y subscript 2 minus y subscript 1 divided by x subscript 2 minus x subscript 1.
The slope of a function is calculated by the change in   y   divided by the change in   x .   It does not matter which coordinate is used as the   ( x 2 , y 2 )   and which is the   ( x 1 , y 1 ) ,   as long as each calculation is started with the elements from the same coordinate pair.

Questions & Answers

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
did you g8ve certificate
Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hllo
John
hi
John
qt rrt
John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
Reply
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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