# 4.1 Linear functions  (Page 5/27)

 Page 5 / 27
$\begin{array}{ccc}\hfill y-4& =& -\frac{3}{4}\left(x-4\right)\hfill \\ \hfill y-4& =& -\frac{3}{4}x+3\hfill \\ \hfill y& =& -\frac{3}{4}x+7\hfill \end{array}$

If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y -axis when the output value is 7. Therefore, $\text{\hspace{0.17em}}b=7.\text{\hspace{0.17em}}$ We now have the initial value $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and the slope $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ so we can substitute $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ into the slope-intercept form of a line.

So the function is $f\left(x\right)=-\frac{3}{4}x+7,$ and the linear equation would be $\text{\hspace{0.17em}}y=-\frac{3}{4}x+7.$

Given the graph of a linear function, write an equation to represent the function.

1. Identify two points on the line.
2. Use the two points to calculate the slope.
3. Determine where the line crosses the y -axis to identify the y -intercept by visual inspection.
4. Substitute the slope and y -intercept into the slope-intercept form of a line equation.

## Writing an equation for a linear function

Write an equation for a linear function given a graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .

Identify two points on the line, such as $\text{\hspace{0.17em}}\left(0,2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,-4\right).\text{\hspace{0.17em}}$ Use the points to calculate the slope.

$\begin{array}{ccc}\hfill m& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{-4-2}{-2-0}\hfill \\ & =& \frac{-6}{-2}\hfill \\ & =& 3\hfill \end{array}$

Substitute the slope and the coordinates of one of the points into the point-slope form.

$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-\left(-4\right)& =& 3\left(x-\left(-2\right)\right)\hfill \\ \hfill y+4& =& 3\left(x+2\right)\hfill \end{array}$

We can use algebra to rewrite the equation in the slope-intercept form.

$\begin{array}{ccc}\hfill y+4& =& 3\left(x+2\right)\hfill \\ \hfill y+4& =& 3x+6\hfill \\ \hfill y& =& 3x+2\hfill \end{array}$

## Writing an equation for a linear cost function

Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are$37.50 per item. Write a linear function $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}C\left(x\right)\text{\hspace{0.17em}}$ is the cost for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items produced in a given month.

The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is$37.50. The variable cost, called the marginal cost, is represented by $\text{\hspace{0.17em}}37.5.\text{\hspace{0.17em}}$ The cost Ben incurs is the sum of these two costs, represented by $\text{\hspace{0.17em}}C\left(x\right)=1250+37.5x.$

## Writing an equation for a linear function given two points

If $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a linear function, with $\text{\hspace{0.17em}}f\left(3\right)=-2,$ and $\text{\hspace{0.17em}}f\left(8\right)=1,$ find an equation for the function in slope-intercept form.

We can write the given points using coordinates.

$\begin{array}{ccc}\hfill f\left(3\right)& =& -2\to \left(3,-2\right)\hfill \\ \hfill f\left(8\right)& =& 1\to \left(8,1\right)\hfill \end{array}$

We can then use the points to calculate the slope.

$\begin{array}{ccc}\hfill m& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{1-\left(-2\right)}{8-3}\hfill \\ & =& \frac{3}{5}\hfill \end{array}$

Substitute the slope and the coordinates of one of the points into the point-slope form.

$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-\left(-2\right)& =& \frac{3}{5}\left(x-3\right)\hfill \end{array}$

We can use algebra to rewrite the equation in the slope-intercept form.

$\begin{array}{ccc}\hfill y+2& =& \frac{3}{5}\left(x-3\right)\hfill \\ \hfill y+2& =& \frac{3}{5}x-\frac{9}{5}\hfill \\ \hfill y& =& \frac{3}{5}x-\frac{19}{5}\hfill \end{array}$

If $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a linear function, with $\text{\hspace{0.17em}}f\left(2\right)=–11,$ and $\text{\hspace{0.17em}}f\left(4\right)=-25,$ write an equation for the function in slope-intercept form.

$y=-7x+3$

## Modeling real-world problems with linear functions

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

Given a linear function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and the initial value and rate of change, evaluate $\text{\hspace{0.17em}}f\left(c\right).$

1. Determine the initial value and the rate of change (slope).
2. Substitute the values into $\text{\hspace{0.17em}}f\left(x\right)=mx+b.$
3. Evaluate the function at $\text{\hspace{0.17em}}x=c.$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar