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# 6 4 Absolute Value And Graphing

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### 6 4 Absolute Value And Graphing

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