1. **Problem Statement:** Find the range and amplitude of the function $$y = \sin(x + 45)$$ and understand its graph shape. 2. **Recall the sine function properties:** - The sine function $$\sin(\theta)$$ has a range of $$[-1, 1]$$. - The amplitude of $$\sin(\theta)$$ is the maximum absolute value of the function, which is $$1$$. - Adding a constant inside the sine function argument, like $$x + 45$$, shifts the graph horizontally but does not change the amplitude or range. 3. **Range:** Since $$y = \sin(x + 45)$$ is just a horizontal shift of $$\sin x$$, the range remains $$[-1, 1]$$. 4. **Amplitude:** The amplitude is the coefficient in front of the sine function. Here, it is $$1$$ (implied), so the amplitude is $$1$$. 5. **Summary:** - Range: $$[-1, 1]$$ - Amplitude: $$1$$ This matches the selected answers in the multiple-choice question. 6. **Graph shape:** The graph of $$y = \sin(x + 45)$$ is the standard sine wave shifted left by 45 units (or $$\frac{\pi}{4}$$ radians), with peaks at $$1$$ and troughs at $$-1$$.