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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

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
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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
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David Reply
what is viscosity?
David
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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
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Adjanou
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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
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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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?
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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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