<< 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.

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.

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.

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.

Practice set a

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

y = 2 3 x + 4
An xy-plane with gridlines, labeled negative five and five and increments of one unit for both axes.

A graph of a line passing through two points with coordinates zero, four and three, two.

y = 3 4 x
An xy-plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates zero, zero and four, three.

Excercises

For the following problems, graph the equations.

y = 2 3 x + 1

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

A graph of a line passing through two points with coordinates zero, one and three, three.

y = 1 4 x 2

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

y = 5 x 4

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

A graph of a line passing through two points with coordinates zero, negative four and one, one.

y = 6 5 x 3

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

y = 3 2 x 5

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

A graph of a line passing through two points with coordinates zero, negative five and two, negative two.

y = 1 5 x + 2

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

y = 8 3 x + 4

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

A graph of a line passing through two points with coordinates zero, four and three, negative four.

y = 10 3 x + 6

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

y = 1 x 4

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

A graph of a line passing through two points with coordinates zero, negative four and one, negative three.

y = 2 x + 1

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

y = x + 2

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

A graph of a line passing through two points with coordinates zero, two and one, three.

y = 3 5 x

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

y = 4 3 x

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

A graph of a line passing through two points with coordinates zero, zero and negative three, four.

y = x

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

y = x

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

A graph of a line passing through two points with coordinates zero, zero and one, negative one.

3 y 2 x = 3

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

6 x + 10 y = 30

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

A graph of a line passing through two points with coordinates zero, three and five, zero.

x + y = 0

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

Excersise for review

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

x 1

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

A horizontal line with arrows on both ends.

( [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.

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

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

m = 2 3

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra i for the community college' conversation and receive update notifications?

Ask