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Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
Mixed
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
Polynomials
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Numbers
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
Polynomials
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Exponents
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
   
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The Cartesian Coordinate System

Suppose we know the sum of two numbers is 6.

We can represent this situation with the equation x + y = 6.

There infinitely many possibilities for x and y. For example:

1 and 5

1 + 5 = 6

-4 and 10

-4 + 10 = 6

0 and 6

0 + 6 = 6

We can visualize this relationship between x and y using two number lines, one for x and the other for y.

First, draw a horizontal number line.

This is usually called the x-axis and is labeled with the variable x.

Next, draw a vertical number line perpendicular to the x-axis.

The two number lines should intersect at their zeros.

The vertical number line is usually called the y-axis and is labeled with the variable y.

The point of intersection, the zero of each number line, is called the origin.

To make it easier to locate a point, we draw a rectangular grid as a background.

The axes and the grid define a flat surface called the xy-plane.

The number lines and grid form a rectangular coordinate system. We typically use x and y for the variables, so a rectangular coordinate system is often called an xy-coordinate system. The French mathematician Rene Descartes (1596-1650) is credited with developing this type of coordinate system, so it is also referred to as the Cartesian coordinate system.

The x- and y-axes divide the plane into four regions called quadrants. We label these with Roman numerals I, II, III, and IV in a counter-clockwise direction beginning in the upper right.

Quadrant

I

II

III

IV

Sign of x

positive

negative

negative

positive

Sign of y

positive

positive

negative

 negative

 

A point on an axis does not lie in a quadrant.

 

Example 1

State the quadrant in which each labeled point lies.

a. A b. B c. C d. D e. E

Solution

a. Quadrant II. Point A lies in the upper left quadrant.

b. Quadrant IV. Point B lies in the lower right quadrant.

c. None. Points on either axis do not lie in a quadrant.

d. Quadrant III. Point D lies in the lower left quadrant.

e. Quadrant I. Point E lies in the upper right quadrant.

 
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