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Complex Conjugates
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Simplified Form of a Square Root
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Determining if a Function has an Inverse
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Using Slopes to Graph Lines
Fractions, Decimals and Percents
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Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
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Factoring a Sum or Difference of Two Cubes
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Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
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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
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Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
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Absolute Value and Distance
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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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