Question 86032
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". This means 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)=x^2}}} Start with the given  function

{{{f(x+3)=(x+3)^2}}}Replace x with {{{x+3}}} to shift the graph 3 units to the left


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,x^2, (x+3)^2),
         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=x^2}}} (red) and the shifted equation {{{y=(x+3)^2}}}(green)

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


Here is the graph


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


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


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