In sine and cosine functions, we have the following forms:
[tex]\begin{gathered} f\mleft(x\mright)=A\sin\mleft(Bx+C\mright)+D \\ f\mleft(x\mright)=A\cos\mleft(Bx+C\mright)+D \end{gathered}[/tex]
Where A is the amplitute, 2π/B is the period, C is the phase shift and D is the vertical shift.
By comparison, we can see that:
[tex]\begin{gathered} f\mleft(x\mright)=A\sin\mleft(Bx+C\mright)+D \\ f\mleft(x\mright)=\sin\mleft(x+45°\mright)+2 \end{gathered}[/tex][tex]\begin{gathered} A=1 \\ B=1 \\ C=45° \\ D=2 \end{gathered}[/tex]
Then, the phase shift is 45°, the vertical shift is 2.
The vertical shift is the same as the middle horizontal axis of the function, so we know that the middle of the function is y = 2. The amplitute is how many units the function varies up and down from the middle. Since the Amplitute is 1, the function varies from 2 - 1 to 2 + 1, that is, it varies from 1 to 3. So, the range of the function is:
[tex]R=\left[1,3\right][/tex]
