<< 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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