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Horizontal Line Test
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Solving Quadratic Equations by Completing the Square
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Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
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Simplified Form of a Square Root
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Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
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Exponential Functions
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The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
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Domain and Range of a Function
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Slope of a Line
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Percent of Change
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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
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
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
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
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 Slope of a Line

Find the slope of the line given two points:

Consider two methods as alternatives for finding the slope: (without using the “slope-formula”)


First, BUILD a TABLE starting with the given points in the middle of the table and find their common differences, dx and dy, then we can use these differences to find the x-sequence and y-sequence.

Since both columns must be arithmetic sequences, to get the next value below for the x-sequence add dx , then add - dx to get the next value above to complete the x-sequence.

Similarly, to get the next value below for the y-sequence add dy, then add - dy to get the next value above, and complete the y-sequence. (Always check both sequences.)

Recall that the slope is and RECORD the value for m.


Plot BOTH points on a grid and move up/(down) and over to FIND values for dy and dx and RECORD the value of m.

Now move using dy and dx to find the next point and continue up/over or down/over to find several points. Label these points and put them in order in a table of sequences.


Example 1:

1. Given (2, -1) and (- 3, 2) plot the points and draw the line through the points.

Put the points in the middle of the table and find the differences dx and dy.

See work below:

We found the slope

NOTE: A “neat” method for finding the slope (without using the “formula”) is shown below:

Ask “how far is it ” from y1 to y2 in which direction (dy)

and “how far is it “ from x1 to x2 in which direction (dx).

Generally, it is better to place the points in the table or chart with x1 < x2 so that dx > 0.

These methods for finding the slope (without using the “formula”) are illustrated in the example below:


Example 2.

Given points: (3, 2), (5, -3) Find Dx and Dy directly from the table or the points.

Either order gives the same slope.

Example 3.

Given points: (3, 2), (6, - 2) Find dy and dx directly from the table or the points.

Either order gives the same slope.

If we are given two points (a, b) and (x, y), and the slope m and we can use the slope- form of the equation for the line:

Multiply each side by (x - a)
We can use this form to find the missing value of two points (a, b), (c, d) if we know three of them.


Example 4.

Find x if the line through (3,3) and (x,9) has the slope of 2.

6 = 2 · (x - 3) Multiply each side by (x - 3)
6 = 2x - 6 Distributive property  
12 = 2x Addition property
6 = x Multiplication property  


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