# 4.1 Linear functions  (Page 2/27)

 Page 2 / 27
$D\left(t\right)=83t+250$

## 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.

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, $\text{\hspace{0.17em}}D\left(t\right)=83t+250,\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(0,250\right)\text{\hspace{0.17em}}$ and represents the distance of the train from the station when it began moving at a constant speed.

Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line $\text{\hspace{0.17em}}f\left(x\right)=2x+1.\text{\hspace{0.17em}}$ 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

$f\left(x\right)=mx+b$

where $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is the initial or starting value of the function (when input, $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ ), and $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ is the constant rate of change, or slope of the function. The y -intercept is at $\text{\hspace{0.17em}}\left(0,b\right).$

## Using a linear function to find the pressure on a diver

The pressure, $\text{\hspace{0.17em}}P,$ in pounds per square inch (PSI) on the diver in [link] depends upon her depth below the water surface, $\text{\hspace{0.17em}}d,$ in feet. This relationship may be modeled by the equation, $\text{\hspace{0.17em}}P\left(d\right)=0.434d+14.696.\text{\hspace{0.17em}}$ Restate this function in words.

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.

## 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) .

how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation