Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:College Algebra

Section 3.1: The Cartesian Coordinate System




Objectives

o The components of the Cartesian coordinate system.o The graph of an equation.o The distance and midpoint formulas.




The Components of the Cartesian Coordinate System

o Many problems are naturally expressed with two or more variables. To solve these problems, we must determine all of the values of the variables that make the equation or inequality true.

o For example, an equation with the variables and will have a solution consisting of a value for and a corresponding value for . A solution of the equation cannot consist of a value for only one of the variables.

o Such equations are graphed on a two-dimensional coordinate system.

xy

x y




The Cartesian Coordinate System

The Cartesian coordinate system, also referred to as the Cartesian plane or the rectangular coordinate system, consists of two perpendicular real number lines (each of which is an axis) intersecting at the point 0 of each line.





The point of intersection is called the origin of the system, and the four quarters defined by the two lines are called the quadrants of the plane, numbered as indicated on the graph to the left.





Because the Cartesian plane consists of two crossed real lines, it is often given the symbol or .

Each point in the plane is identified by a unique pair of numbers called an ordered pair.

¡ ¡2¡

P





In a given ordered pair the first number (the first coordinate) indicates the horizontal displacement of the point from the origin, and the second number (the second coordinate) indicates the vertical displacement.





Caution!Mathematics uses parentheses to denote ordered pairs as well as open intervals, which sometimes leads to confusion. You must rely on the context to determine the meaning of any parentheses you encounter. For instance, in the context of solving a one-variable inequality, the notation most likely refers to the open interval with endpoints at and where as in the context of solving an equation in two variables, probably refers to a point in the Cartesian plane.

2,52

5 2,5




The Graph of an Equation

The horizontal number line is referred to as the x-axis, the vertical number line as the y-axis, and the two coordinates of the ordered pair as the x-coordinate and the y-coordinate.

,a b




The Graph of an Equation

o The graph of an equation consists of a depiction in the Cartesian plane of all of those ordered pairs that make up the solution set of the equation.

o We can find individual ordered pair solutions of a given equation by selecting numbers that seem appropriate for one of the variables and then solving the equation for the other variable.




Example 1: The Graph of an Equation

Sketch the graph of the following equation by plotting points. 2 5 10x y x y

3

00

5

1

165

2

5352

85





Sketch graphs of the following equations by plotting points.2 2 6 0x y x x y

01

2

3

4

56

05 2.2

2 2 2.8

3

2 2 2.8

5 2.2 0





Sketch a graph of the following equation by plotting points.2 2y x x x y

01

1324

0,21

33

88




The Distance Formula

Let and be the coordinates of two arbitrary points in the plane. By drawing the dotted lines parallel to the coordinate axes, we can form a right triangle. Note that we are able to determine the coordinates of the vertex at the right angle from the two vertices and .

1 1,x y 2 2,x y

1 1,x y 2 2,x yy

x

2 22 1 2 1d x x y y

2 1x x

2 1y y

1 1,x y

2 2,x y

2 1,x y





o The lengths of the two perpendicular sides of the triangle from the previous slide are easily calculated, as these lengths correspond to distances between numbers on the real number lines.

o We can apply the Pythagorean Theorem to determine the distance, as labeled on the previous slide.

2 2 2a b c

2 222 1 2 1 , sod x x y y

2 22 1 2 1d x x y y





Letting and represent two points on the Cartesian plane, the distance between these two points may be found using the following distance formula, derived from the Pythagorean Theorem:

1 1,x y 2 2,x y

2 22 1 2 1d x x y y




Example 4: The Distance Formula

Determine the distance between and . 1,4 3,7

2 22 1 2 1d x x y y

2 23 1 7 4d

16 9d

25d

5d




The Midpoint Formula

Consider the points plotted below. The of the midpoint should be the average of the two of the given points, and similarly for the .

-coordinatex'sx

-coordinateyy

x

1 1,x y

2 2,x y

2 1,x y

1 2 1 2,2 2

x x y y 1 2

2 ,2

y yx

1 21,

2x x y




The Midpoint Formula

Letting and represent two points on the Cartesian plane, the midpoint between these two points may be found using the following midpoint formula, which finds the average of the two values and the average of the two values.

1 1,x y 2 2,x y

-x-y

1 2 1 2,2 2

x x y y




Example 5: The Midpoint Formula

Determine the midpoint of the line segment joining and .

1,4 3,7

1 2 1 2,2 2

x x y y

1 3 4 7,2 2

1,5.5

Hawkes Learning Systems: College Algebra

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