Step 1
The parent function f(x) is given as;
[tex]f(x)=3\sin (x)+1[/tex]
If we transform the function by adding 1 to it we will have;
[tex]\begin{gathered} f(x)=3\sin (x)+1+1 \\ f(x)=3\sin (x)+2 \end{gathered}[/tex]
We have the following graph;
which means when you add 1 to the to get f(x)=3sin(x)+2, the function is shifted up by 1 unit.
Step 2
If the function is further transformed to;
[tex]f(x)=3\sin (\frac{x}{4})+1[/tex]
we will have the graph below;
This means that the graph stretches horizontally by a factor of 4.
Therefore the changes f(x) passes through to g(x) are;
[tex]\begin{gathered} f(x)=2\sin (\frac{x}{4})+1_{}--(A\text{ horizontal stretch by a factor of 4)} \\ g(x)=2\sin (\frac{x}{4})+2---(A\text{ shift up by 1 unit)} \end{gathered}[/tex]
Answer; The graph is stretched horizontally by a factor of 4 and shifted up by 1 unit.
