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In this section you will:
  • Plot ordered pairs in a Cartesian coordinate system.
  • Graph equations by plotting points.
  • Graph equations with a graphing utility.
  • Find x -intercepts and y -intercepts.
  • Use the distance formula.
  • Use the midpoint formula.
Road map of a city with street names on an x, y coordinate grid. Various points are marked in red on the grid lines indicating different locations on the map.

Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in [link] . Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.

Plotting ordered pairs in the cartesian coordinate system

An old story describes how seventeenth-century philosopher/mathematician René Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.

While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system    , a grid system having perpendicular axes. Descartes named the horizontal axis the x- axis and the vertical axis the y- axis .

The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x -axis and the y -axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant    ; the quadrants are numbered counterclockwise as shown in [link]

This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I.  The upper left section is labeled: Quadrant II.  The lower left section is labeled: Quadrant III.  The lower right section is labeled: Quadrant IV.

The center of the plane is the point at which the two axes cross. It is known as the origin    , or point ( 0 , 0 ) . From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the x- axis and up the y- axis; decreasing, negative numbers to the left on the x- axis and down the y- axis. The axes extend to positive and negative infinity as shown by the arrowheads in [link] .

This is an image of an x, y coordinate plane.  The x and y axis range from negative 5 to 5.

Each point in the plane is identified by its x- coordinate    , or horizontal displacement from the origin, and its y- coordinate    , or vertical displacement from the origin. Together, we write them as an ordered pair    indicating the combined distance from the origin in the form ( x , y ) . An ordered pair is also known as a coordinate pair because it consists of x- and y -coordinates. For example, we can represent the point ( 3 , −1 ) in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See [link] .

This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.  The point (3, -1) is labeled.  An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.

When dividing the axes into equally spaced increments, note that the x- axis may be considered separately from the y- axis. In other words, while the x- axis may be divided and labeled according to consecutive integers, the y- axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.

Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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