Question 155777

Remember, if {{{abs(x)=A}}}, then {{{x=A}}} or {{{x=-A}}}. So this means that {{{abs(x-10)}}} can be broken up into {{{(x-10)}}} and {{{-(x-10)=-x+10}}}


{{{y=-abs(x-10)+2}}} Start with the given equation



{{{y=-(x-10)+2}}} Replace {{{abs(x-10)}}} with {{{(x-10)}}} (which is the first part)



{{{y=-x+10+2}}} Distribute the negative.



{{{y=-x+12}}} Combine like terms.



Now graph the equation {{{y=-x+12}}} (note: let me know if you help graphing this)



{{{graph(500,500,-5,15,-8,12,-x+12)}}} Graph of {{{y=-x+12}}}



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{{{y=-abs(x-10)+2}}} Go back to the given equation



{{{y=-(-x+10)+2}}} Replace {{{abs(x-10)}}} with {{{-x+10}}} (which is the second part) 



{{{y=x-10+2}}} Distribute the negative



{{{y=x-8}}} Combine like terms.



Now graph the equation {{{y=x-8}}} 



{{{graph(500,500,-5,15,-8,12,x-8)}}} Graph of {{{y=x-8}}}



Now graph the two equations together:



{{{graph(500,500,-5,15,-8,12,-x+12,x-8)}}} Graph of {{{y=-x+12}}} (red) and graph of {{{y=x-8}}} (green)


Because the sign in front of the absolute value is a negative sign, this tells us that we have to erase the portions of the graph that are above the intersection like this:



{{{graph(500,500,-5,15,-8,12,-abs(x-10)+2)}}} Graph of {{{y=-abs(x-10)+2}}}



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Answer:


So the graph of {{{y=-abs(x-10)+2}}} is 


{{{graph(500,500,-5,15,-8,12,-abs(x-10)+2)}}}