<< Chapter < Page Chapter >> Page >
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information.The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module contains an overview of the chapter "Graphing Linear Equations and Inequalities in One and Two Variables".

Overview

  • Using the Slope and Intercept to Graph a Line

Using the slope and intercept to graph a line

When a linear equation is given in the general form , a x + b y = c , we observed that an efficient graphical approach was the intercept method. We let x = 0 and computed the corresponding value of y , then let y = 0 and computed the corresponding value of x .

When an equation is written in the slope-intercept form , y = m x + b , there are also efficient ways of constructing the graph. One way, but less efficient, is to choose two or three x -values and compute to find the corresponding y -values . However, computations are tedious, time consuming, and can lead to errors. Another way, the method listed below, makes use of the slope and the y -intercept for graphing the line. It is quick, simple, and involves no computations.

    Graphing method

  1. Plot the y -intercept ( 0 , b ) .
  2. Determine another point by using the slope m .
  3. Draw a line through the two points.

Recall that we defined the slope m as the ratio y 2 y 1 x 2 x 1 . The numerator y 2 y 1 represents the number of units that y changes and the denominator x 2 x 1 represents the number of units that x changes. Suppose m = p q . Then p is the number of units that y changes and q is the number of units that x changes. Since these changes occur simultaneously, start with your pencil at the y -intercept , move p units in the appropriate vertical direction, and then move q units in the appropriate horizontal direction. Mark a point at this location.

Sample set a

Graph the following lines.

y = 3 4 x + 2

  1. The y -intercept is the point ( 0 , 2 ) . Thus the line crosses the y -axis 2 units above the origin. Mark a point at ( 0 , 2 ) .

     An xy coordinate plane with gridlines from negative five to five in increments of one unit for both axes. The point zero, two is plotted and labeled on the grid.
  2. The slope, m , is 3 4 . This means that if we start at any point on the line and move our pencil 3 units up and then 4 units to the right, we’ll be back on the line. Start at a known point, the y -intercept ( 0 , 2 ) . Move up 3 units, then move 4 units to the right. Mark a point at this location. (Note also that 3 4 = 3 4 . This means that if we start at any point on the line and move our pencil 3 units down and 4 units to the left , we’ll be back on the line. Note also that 3 4 = 3 4 1 . This means that if we start at any point on the line and move to the right 1 unit, we’ll have to move up 3 / 4 unit to get back on the line.)

    Starting at point with coordinates zero, two move three units up and four units right to reach to the point with coordinates four, five.
  3. Draw a line through both points.

    A graph of a line passing through two points with coordinates zero, two, and four, five.
Got questions? Get instant answers now!

y = 1 2 x + 7 2

  1. The y -intercept is the point ( 0 , 7 2 ) . Thus the line crosses the y -axis 7 2 units above the origin. Mark a point at ( 0 , 7 2 ) , or ( 0 , 3 1 2 ) .

    An xy coordinate plane with gridlines from negative five to five and increments of one unit for both axes. The point zero, three and one half is plotted and labeled.
  2. The slope, m , is 1 2 . We can write 1 2 as 1 2 . Thus, we start at a known point, the y -intercept ( 0 , 3 1 2 ) , move down one unit (because of the 1 ), then move right 2 units. Mark a point at this location.

    Starting at point with coordinates zero, three and half move one unit downward and two units right to reach to the point with coordinates two, two and half.
  3. Draw a line through both points.

    A graph of a line passing through two points with coordinates zero, three and one half; and two, two and one half.
Got questions? Get instant answers now!

y = 2 5 x

  1. We can put this equation into explicit slope-intercept by writing it as y = 2 5 x + 0 .

    The y -intercept is the point ( 0 , 0 ) , the origin. This line goes right through the origin.

    An xy coordinate plane with gridlines from negative five to five and increments of one unit for both axes. The origin is labeled with the coordinate pair zero, zero.
  2. The slope, m , is 2 5 . Starting at the origin, we move up 2 units, then move to the right 5 units. Mark a point at this location.

    A graph of a line passing through two points with coordinates zero, zero; and five, two. Starting at a point with coordinates zero, zero moves two units up and five units to the right to reach to the point with coordinates five, two.
  3. Draw a line through the two points.
Got questions? Get instant answers now!

y = 2 x 4

  1. The y -intercept is the point ( 0 , 4 ) . Thus the line crosses the y -axis 4 units below the origin. Mark a point at ( 0 , 4 ) .

    A point with the coordinates zero, negative four plotted in an xy plane.
  2. The slope, m , is 2. If we write the slope as a fraction, 2 = 2 1 , we can read how to make the changes. Start at the known point ( 0 , 4 ) , move up 2 units, then move right 1 unit. Mark a point at this location.

    A graph of a line passing through two points with coordinates zero, negative four and one, negative two.
  3. Draw a line through the two points.
Got questions? Get instant answers now!

Practice set a

Use the y -intercept and the slope to graph each line.

Excercises

For the following problems, graph the equations.

Excersise for review

( [link] ) Solve the inequality 2 4 x x 3 .

x 1

Got questions? Get instant answers now!

( [link] ) Graph the inequality y + 3 > 1 .

A horizontal line with arrows on both ends.

Got questions? Get instant answers now!

( [link] ) Graph the equation y = 2 .

An xy-plane with gridlines, labeled negative five and five on the both axes.

A graph of a line parallel to x-axis in an xy plane.The line crosses the y-axis at y equals negative two.

Got questions? Get instant answers now!

( [link] ) Determine the slope and y -intercept of the line 4 y 3 x = 16 .

Got questions? Get instant answers now!

( [link] ) Find the slope of the line passing through the points ( 1 , 5 ) and ( 2 , 3 ) .

m = 2 3

Got questions? Get instant answers now!

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask