Question

G=f(x+3)+2 I needs help remembering which one is up and down and which one is left to right

Answer 1

If "a" is a positive number, then we can perform the following transformations:

1) A shift to the left (or to the right)

2) A shift up (or down)

1)

This transformation can be expressed as:

$\begin{gathered} f\mleft(x\mright)\rightarrow f\mleft(x+a\mright)\text{ \lparen Shift a units to the left\rparen} \\ f\mleft(x\mright)\operatorname{\rightarrow}f\mleft(x-a\mright)\operatorname{\lparen}\text{Shift a units to the right}\operatorname{\rparen} \end{gathered}$

2)

This transformation can be expressed as:

$\begin{gathered} f\mleft(x\mright)\rightarrow f\lparen x)+a\text{ \lparen Shift a units up\rparen} \\ f\mleft(x\mright)\operatorname{\rightarrow}f\lparen x)-a\operatorname{\lparen}\text{Shift a units down}\operatorname{\rparen} \end{gathered}$

From this, if we have:

$G\left(x\right)=f\mleft(x+3\mright)+2$

Then, we have:

$\begin{gathered} f\mleft(x\mright)\rightarrow f\mleft(x+3\mright)\text{ \lparen A shift 3 units to the left\rparen} \\ f\mleft(x\mright)\rightarrow f\mleft(x\mright)+2\text{ \lparen A shift 2 units up\rparen} \end{gathered}$

If we combine both transformations, we have a final graph of G(x) that is shifted 3 units to the left and 2 units up with respect to the graph of f(x)