Question 86030
If you want to shift any function, here is the general rule:


{{{f(x-a)+k}}} where a represents the units shifted horizontally and k represents the units shifted veritcally


note: notice the negative "a". If you have {{{f(x-3)}}}, it means shift 3 units in the <b>positive</b> direction. If you have {{{f(x+3)}}}, it means shift 3 units in the <b>negative</b> direction. So just remember that the negative is telling you to shift in the <b>opposite</b> direction.



{{{f(x)=abs(x)}}} Start with the given  function

{{{f(x-3)=abs(x-3)}}}Replace x with {{{x-3}}} to shift the graph 3 units to the right


Here is the graph

{{{drawing( 500, 500, -5, 5, -5, 5,
         grid(1),
         arrow(0,0,3,0),
         graph( 500, 500, -5, 5, -5, 5,abs(x), abs(x-3)),
         circle(3,0,0.05),
         circle(3,0,0.08),
         circle(0,0,0.05),
         circle(0,0,0.08)
)}}}graph of the original equation {{{y=abs(x)}}} (red) and the shifted equation {{{y=abs(x-3)}}}(green)

{{{f(x-3)-2=abs(x-3)-2}}} Now subtract 2 from the whole equation to shift the graph down 2 units


Here is the graph


{{{drawing( 500, 500, -5, 5, -5, 5,
         grid(1),
         arrow(0,0,3,-2),
         graph( 500, 500, -5, 5, -5, 5,abs(x), abs(x-3)-2),
         circle(3,-2,0.05),
         circle(3,-2,0.08),
         circle(0,0,0.05),
         circle(0,0,0.08)
)}}}graph of the original equation {{{y=abs(x)}}} (red) and the shifted equation {{{y=abs(x-3)-2}}}(green)


So in short, to shift the graph 3 units to the right and shift the graph 2 units down,you simply use this composite function:


{{{g(x)=f(x-3)-2}}} where {{{g(x)}}} is your new function