# 2.1 The rectangular coordinate systems and graphs

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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.

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]

The center of the plane is the point at which the two axes cross. It is known as the origin    , or point $\left(0,0\right).$ 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] .

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 $\text{\hspace{0.17em}}\left(x,y\right).\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(3,-1\right)\text{\hspace{0.17em}}$ 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] .

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.

sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
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stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
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what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb