4.1 Linear functions (Page 4/27)

College algebra

Page 4 / 27

Q&A

Are the units for slope always $\frac{units for the output}{units for the input} ?$

Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.

A General Note

Calculate slope

The slope, or rate of change, of a function $m$ can be calculated according to the following:

m = \frac{change in output (rise)}{change in input (run)} = \frac{Δ y}{Δ x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

where $x_{1}$ and $x_{2}$ are input values, $y_{1}$ and $y_{2}$ are output values.

How To

Given two points from a linear function, calculate and interpret the slope.

Determine the units for output and input values.
Calculate the change of output values and change of input values.
Interpret the slope as the change in output values per unit of the input value.

Finding the slope of a linear function

If $f (x)$ is a linear function, and $(3, −2)$ and $(8, 1)$ are points on the line, find the slope. Is this function increasing or decreasing?

The coordinate pairs are $(3, −2)$ and $(8, 1) .$ To find the rate of change, we divide the change in output by the change in input.

m = \frac{change in output}{change in input} = \frac{1 - (−2)}{8 - 3} = \frac{3}{5}

We could also write the slope as $m = 0.6.$ The function is increasing because $m > 0.$

Got questions? Get instant answers now!

Try It

If $f (x)$ is a linear function, and $(2, 3)$ and $(0, 4)$ are points on the line, find the slope. Is this function increasing or decreasing?

$m = \frac{4 - 3}{0 - 2} = \frac{1}{- 2} = - \frac{1}{2};$ decreasing because $m < 0.$

Got questions? Get instant answers now!

Finding the population change from a linear function

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The rate of change relates the change in population to the change in time. The population increased by $27, 800 - 23, 400 = 4400$ people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.

\frac{4,400 people}{4 years} = 1,100 \frac{people}{year}

So the population increased by 1,100 people per year.

Got questions? Get instant answers now!

Try It

The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

m = \frac{1, 868 - 1, 442}{2, 012 - 2, 009} = \frac{426}{3} = 142 people per year

Got questions? Get instant answers now!

Writing and interpreting an equation for a linear function

Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form . Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function $f$ in [link] .

This graph shows a linear function graphed on an x y coordinate plane. The x axis is labeled from negative 2 to 8 and the y axis is labeled from negative 1 to 8. The function f is graph along the points (0, 7) and (4, 4).

We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose $(0, 7)$ and $(4, 4) .$

\begin{matrix} m & = & \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ = & \frac{4 - 7}{4 - 0} \\ = & - \frac{3}{4} \end{matrix}

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.

\begin{matrix} y - y_{1} & = & m (x - x_{1}) \\ y - 4 & = & - \frac{3}{4} (x - 4) \end{matrix}

If we want to rewrite the equation in the slope-intercept form, we would find

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	9 Modeling real-world problems with linear functions By OpenStax Read Online Course
	7 Interpreting slope as a rate of change By OpenStax Read Online Course
	10 Lec:10 Sampling Confidence Intervals By Janet Forrester Start Quiz
	2 Sociology 02 Sociological Research MCQ By OpenStax Start Quiz
©flickr: Rod	Medieval Africa By Jesenia Wofford Start Quiz
	6 Psychology MCQ 2010 2 Exam By John Gabrieli Start Exam
©flickr: Dave	Infection Control By Cath Yu Start Quiz
	6 Enterprise JavaBeans By JavaChamp Team Start Quiz
	2 CDL Quiz - Air Brakes Endorsement Part 2 By Jazzycazz Jackson Start Quiz
	Real Estate Finance Exam 2006 By David Geltner Start Exam
©flickr: eLife	Mycobacterial Infections (Mandell 8th Ch 251-254) By Cath Yu Start Quiz
	Social Organization Kinship By Richley Crapo Start Assignment