6 4 absolute value and graphing
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Absolute Value and Absolute Value and GraphingGraphing
Review of Chapter 6.4Pages 295-297
What’s the Deal?What’s the Deal?
• In this lesson – We will review domain and range.– We will graph the results of how
absolute value affects variables.
y = +7y = +7
• Since every y value equals 7, we graph with a zero slope.
• y = (0)x + 7• Using an x-y box
(-4, +7)(-2, +7)( 0, +7)(+2, +7)(+4, +7)
x y-4 7-2 7 0 7+2 7+4 7
y = +7y = +7
• Arrows– Show that all points
beyond also make the equation true.
• Using an x-y box(-100, +7)(-52, +7)( 10, +7)(+20, +7)(+144, +7)
What are the domain and What are the domain and range?range?
• The domain for an equation is all the values that will work for x.
• The range for an equation is all the values that will work for y.
• Domain: {all real numbers}
• Range: {+7}
x y-4 7-2 7 0 7+2 7+4 7
7y
Number TermsNumber Terms
• Integers {…,-6,-5,-4,-3,-2,-1,0,1,2,3,4…}
• Whole Numbers { 0,1,2,3,4,5,6,7…}
• Counting Numbers { 1,2,3,4,5,6,7…}
• Real Numbers {integers, fractions, decimal numbers, repeating decimals, non-repeating decimals….}
Task: Graph Task: Graph yy = 2 = 2xx-1 and find -1 and find the domain & rangethe domain & range
• Once again use and x-y box. (y=mx+b)
• Fill in -4 for x.y=2(-4)-1y=-8-1y= -9
Do the same for the rest of the values chosen.
x y-4-2 0+2+4
-9
When you are finished, go to the next slide.
Graph the pointsGraph the points
Add a line
x y-4 -9-2 -5 0 -1+2 3+4 7
Name the domain and Name the domain and range.range.
Any number can be used as x or y.
Domain:{all real numbers}
Range:{all real numbers}
x y-4 -9-2 -5 0 -1+2 3+4 7
Graph y = | x-2 |Graph y = | x-2 |
• Start by using an x-y box with 0 and some negative and positive numbers for x.|-5-2| = |-7||-7| = 7
+7
x y-5-1 0+2+6+8
Graph y = | x-2 |Graph y = | x-2 |
• Show the graphed pairs.
• Fill in a few more values that work. x y
-5 7-1 3 0 2+2 0+6 4+8 6
IsIs y = | x-2 | y = | x-2 | a linear a linear equation?equation?
• You can begin to see that the values form a V when graphed, not a line.
Any real number can be used as x, but no negative numbers are used for y.
Domain:{all real numbers}Range:{all wholel
numbers}
How is y = - | x-2 | How is y = - | x-2 | different?different?
• All the y values are opposite the previous equation’s y-values. x y
-5 7-1 3 0 2+2 0+6 4+8 6
x y-5 -7-1 -3 0 -2+2 -0+6 -4+8 -6
Absolute Value Equations Absolute Value Equations with Inequalitieswith Inequalities
Key: Split the equation into two parts, a positive
and negative side.
Absolute ValueAbsolute Value
• To find Absolute value,– find the solution
inside the absolute value signs
– Make that value positive (+)
– Continue on with order of operations outside the signs
• Example:
7 | 2 3 | 7 5
Making Use of Absolute Making Use of Absolute ValueValue
• Adding a positive to a negative integer– Which has the higher
absolute value?– The positive or
negative sign of that number is in the answer.
– Now find the difference.
14 ( 27) - 13
Find the value: |x-2| =7Find the value: |x-2| =7
• This has two possible answers.
• There must be a handy pattern to use to find both.
• |+9-2| =7• |-5-2| =7
How to find the value: |x-2| How to find the value: |x-2| =7=7
• This problem should be done twice.
• Procedure:– Remove the absolute value signs– Solve for the positive answer.– Rewrite without absolute value
signs.– Solve for negative answer.
Procedure |x-2| =7Procedure |x-2| =7
• Remove absolute value signs.
x - 2 = 7• Solve for x
x +2 -2 = +2 + 7
x = 9
• Make 2nd equation’s answer negative.
x - 2 = -7• Solve for x
x +2 -2 = +2 - 7x = -5
Let’s take another look at a previous slide and see if the answers given were correct.
Find the value: |x-2| =7Find the value: |x-2| =7
• This has two possible answers.
• There must be a handy pattern to use to find both.
|+9-2| =7|-5-2| =7
x = -5 OR +9Give both answers.
Procedure for |Procedure for |x-10x-10| =4.5| =4.5
• Remove absolute value signs.
x - 10 = 4.5• Solve for x
x +10 -10 = +10 + 4.5
x = 14.5
• Make 2nd equation’s answer negative.
x - 10 = -4.5• Solve for x
x +10 -10 = +10 – 4.5
x = -5.5
x = -5.5 OR +14.5
Solve for | 2Solve for | 2x-14 x-14 | = 8| = 8
• Part One. 2x - 14 = 8
+14 +14 2x +0 = 22
x = +11
• Part Two. 2x - 14 = -8
+14 +14
2x +0 = 6
x = 3 x = +3 OR +11
2 22
2 2
x 2 6
2 2
x
Solve for |x - (-5)| Solve for |x - (-5)| 8 8
• Part One.
x + 5 8 -5 -5 x +0 3
x +3
• Switch the sign for the negative. Why?
x + 5 -8 -5 -5
x +0 -13
x -13 x -13 OR x
+3
Graph the solution forGraph the solution for |x - (-5)| |x - (-5)| 8 8
• You can rewrite the OR statement.
• Then graph.
• x -13 OR x +3
• -13 x +3
-6 -4 -2 0 +2 +4 +6
Graph the solution to the Graph the solution to the equation.equation.
-14 -12 -10 -8 -6 +4 -2 0 +2
13 3x
Solve for |x - 6| Solve for |x - 6| >> 5 5
• Part One.
x - 6 > 5 +6 +6 x +0 > 11
x > +11
• Switch the sign for the negative. Why?
x - 6 < -5 +6 +6
x +0 < +1
x < +1 x > +1 OR x <
+11
Graph the solution to Graph the solution to |x - 6| |x - 6| >> 5 5
-4 -2 0 2 4 6 8 10 12
x > +1 OR x < +11
Extras for presentationExtras for presentation
x y-4-2 0+2+4
-6 -4 -2 0 +2 +4 +6