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Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Inequalities
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Polynomials
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Conjugates
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Formulas
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Roots
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
 
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The Rectangular Coordinate System

Keep tthe following in mind:

(i) When the coordinates of a point are written as a bracketed pair of numbers, we always write the horizontal coordinate first and the vertical coordinate second. This is why (a, b) is often called an ordered pair. For example, the point (5, 2) is at a quite different location than the point (2, 5):

(ii) Since the identity of a point comes from its measured position or location with respect to the scales on the coordinate axes, it is mandatory to show the scales on the axes explicitly , and to label the axes explicitly. If you omit the scale markings and labels, you end up with a meaningless graph.

(iii) The graph of a single point is just a dot at the appropriate location. No additional lines, etc. should be drawn unless they are requested.

(iv) Points on the horizontal axis have coordinates of the form (b, 0) – that is, their vertical coordinates are equal to zero. Points on the vertical axis have coordinates of the form (0, b) – that is, their horizontal coordinate is zero. The origin has coordinates (0, 0) – both of its coordinates are zero.

(v) The rectangular coordinate axes divide the plane into four regions, called quadrants . The quadrants are identified by number, with quadrant 1, or the first quadrant being the upper right one. They are arranged as shown:

 

Plotting Graphs of Formulas

To graph a formula by hand, the usual procedure is to

i) make a table of x and y values for a representative collection of values of x in the specified interval

ii) plot these pairs of values as points on a rectangular coordinate system

iii) join the points by a line or smooth curve

 

Example:

Plot the graph of y = 2x + 3 for x between -3 and +3 inclusive.

Solution:

(i) start by making a table of coordinates of representative points:

(ii) we need coordinate axes that have x going from -3 to +3 and y going from -3 to +9. Set these up and then plot the points as dots at the appropriate locations:

(iii) The points appear to lie on a straight line. Laying a straight edge on the graph confirms this, so in this case, just draw a straight line through the points to complete the graph, as shown above. Had we recognized that the formula, y = 2x + 3, is the type of formula that gives a straight line graph, we could have saved ourselves some work, since we would need to plot only two points to get the entire graph.

 

Example:

The area, A, of a square with sides of length s is given by the formula

A = s 2

Plot a graph of A vs s for s = 0 through s = 5.

Solution:

This example illustrates several issues:

(i) People often use the symbols x and y generically when speaking about graphs. However, we can create a rectangular coordinate system for any pair of symbols or variables we wish to use.

(ii) When asked to plot the graph of “A vs. s”, we are to make the vertical axis the A-axis, and the horizontal axis is the s-axis:

So, here, we start again by making a table of coordinates of representative points for the graph:

Now, plot the points on a rectangular coordinate system which has a horizontal (s) axis running from s = 0 to s = 5, at least, and a vertical (A) axis running at least from A = 0 to A = 25.

We see in this case that the points appear to follow a curved path, bending upwards as you move towards the right. The graph starts out with quite a shallow slope around s = 0, but appears to get steeper and steeper as we move towards higher values of s.

 
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