<< Chapter < Page Chapter >> Page >
D ( t ) = 83 t + 250

Representing a linear function in tabular form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in [link] . From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

Table with the first row, labeled t, containing the seconds from 0, 1, 2, 3, and with the second row, labeled D (t), containing the meters 250, 333, 416, and 499. Each value in the first row increases by 1 second, and each value in the second row increases by 83 meters.
Tabular representation of the function D showing selected input and output values

Can the input in the previous example be any real number?

No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.

Representing a linear function in graphical form

Another way to represent linear functions is visually, using a graph. We can use the function relationship from above, D ( t ) = 83 t + 250 , to draw a graph as represented in [link] . Notice the graph is a line . When we plot a linear function, the graph is always a line.

The rate of change, which is constant, determines the slant, or slope    of the line. The point at which the input value is zero is the vertical intercept, or y -intercept    , of the line. We can see from the graph that the y -intercept in the train example we just saw is ( 0 , 250 ) and represents the distance of the train from the station when it began moving at a constant speed.

This is a graph with y-axis labeled “Distance (m)” and x-axis labeled “Time (s).” The x-axis spans from 0 to 5 and is marked in increments of one. The y-axis spans from 0 to 500 and is marked in increments of one hundred. The graph shows an increasing function. As time increases, distance also increases. The line is graphed along the points (0, 250) and (1, 333)
The graph of D ( t ) = 83 t + 250 . Graphs of linear functions are lines because the rate of change is constant.

Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line f ( x ) = 2 x + 1. Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.

Linear function

A linear function    is a function whose graph is a line. Linear functions can be written in the slope-intercept form    of a line

f ( x ) = m x + b

where b is the initial or starting value of the function (when input, x = 0 ), and m is the constant rate of change, or slope of the function. The y -intercept is at ( 0 , b ) .

Using a linear function to find the pressure on a diver

The pressure, P , in pounds per square inch (PSI) on the diver in [link] depends upon her depth below the water surface, d , in feet. This relationship may be modeled by the equation, P ( d ) = 0.434 d + 14.696. Restate this function in words.

This figure shows a scuba diver.
(credit: Ilse Reijs and Jan-Noud Hutten)

To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.

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

Determining whether a linear function is increasing, decreasing, or constant

The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function    , as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in [link] (a) . For a decreasing function    , the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in [link] (b) . If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in [link] (c) .

Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin

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