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PREC11 Unit6 notes - PRE-CALCULUS 11 Unit 6 Day 1 ABSOLUTE...

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PRE-CALCULUS 11 Unit 6 – Day 1: ABSOLUTE VALUE ABSOLUTE VALUE The absolute value of a real number a is the distance from a to 0 on a number line ; an absolute value cannot be negative. The absolute value of a is written as | a | . The absolute value of 7 is written as | 7 | , the distance from 7 to 0, | 7 | = 7 The absolute value of - 5 is written as | - 5 | , the distance from - 5 to 0, | - 5 | = 5 The absolute value of 0 is written as | 0 | , the distance from 0 to 0, | 0 | = 0 Sometimes the absolute value does not change what is between the vertical lines; sometimes the absolute value changes what is between the vertical lines to its opposite. | a | = a when a is | a | = - a when a is The Mathematical Definition for Absolute Value | x | = < = - 0 if 0 if 0 if 0 x x x x x or | x | = x x x x - < if 0 if 0 | - x | = Does | 6 - 4 | equal 6 - 4 ? Why? Does | 4 - 6 | equal 4 - 6 ? Why? Does | 4 - 6 | equal - (4 - 6) ? | x - b | = x - b when | x - b | = - ( x - b ) when 0
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Unit 6: Day 1 notes - Absolute Value Page 2 of 2 Absolute value can be used to ensure an expression will not be negative. Does | 6 - 4 | equal | 4 - 6 | ? Does | x - b | always equal | b - x | ? example: Write an expression that will be the difference of any two numbers m and n . The difference between two numbers is not negative. Since we do not know which number is the larger, subtract in any order and take the absolute value of the subtraction. | m - n | or | n - m | The absolute value of a sum or difference should not be written as the sum or difference of separate absolute values. examples: Evaluate a) | 4 + - 6 | b) | 4 | + | - 6 | c) | 4 | - | 6 | Does | 4 + - 6 | equal 4 + - 6 ? Does | 4 + - 6 | equal | 4 | + | - 6 | ? Does | x + y | equal | x | + | y | for all values for x and y ? The absolute value of a product or quotient can be written as the product or quotient of separate absolute values. examples: Evaluate a) | ( - 5)(7) | b) | - 5 | | 7 | c) - | (5)(7) | Does | ( - 5)(7) | equal ( - 5)(7) ? Does | ( - 5)(7) | equal | ( - 5) | | (7) | ? Does | x y | equal | x | | y | for all values for x and y ? Recall: 2 x = x when x 0 and 2 x = - x when x < 0 | x | = x when x 0 and | x | = - x when x < 0 Therefore: 2 x = | x |
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PRE-CALCULUS 11 Unit 6 – Day 2: ABSOLUTE VALUE FUNCTIONS ABSOLUTE VALUE FUNCTIONS An absolute value function is a function that includes an absolute value of a variable expression. The basic absolute value function is y = | x | . We will be drawing the graph of functions that are the absolute value of linear functions or the absolute value of quadratic functions by transforming the graph of the linear or quadratic function’s graph. GRAPHING y = | m x + b | , THE ABSOLUTE VALUE OF A LINEAR FUNCTION example: Draw the graph of y = | x + 2 | by transforming the graph of y = x + 2 . x x + 2 = y - 5 - 3 ( - 5, - 3) - 4 - 2 ( - 4, - 2) - 3 - 1 ( - 3, - 1) - 2 0 ( - 2,0) - 1 1 ( - 1,1) 0 2 (0,2) 1 3 (1,3) x | x + 2 | = y - 5 3 ( - 5,3) - 4 2 ( - 4,2) - 3 1 ( - 3,1) - 2 0 ( - 2,0) - 1 1 ( - 1,1) 0 2 (0,2) 1 3 (1,3) When absolute value is applied to a function, the original function's graph is transformed - all points below the x -axis (points with negative y -coordinates) are reflected about the x -axis to above the x -axis .
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