Transforming sine and cosine graphs can be tricky and make things confusing for students. Here are some important ways these transformations change the graphs:

1. Vertical Shifts: When we add a number $c$ to the function, like in $y = \sin(x) + c$ or $y = \cos(x) + c$, the graph moves up or down. This part seems simple, but many students forget whether to move up or down, depending on if $c$ is positive (up) or negative (down).

2. Horizontal Shifts: When we use the transformation $y = \sin(x - d)$ or $y = \cos(x - d)$, the graph shifts left or right. Students sometimes make mistakes in figuring out how far to shift and in which direction, so they need to pay close attention to that change.

3. Amplitude Changes: Changing the function to $y = A\sin(x)$ or $y = A\cos(x)$ affects how tall or short the waves are—that's called the amplitude. Students might forget what $A$ does, especially if $A$ is less than 1 or a negative number, which can cause the waves to flip or shrink.

4. Period Adjustments: When we change the function to $y = \sin(bx)$ or $y = \cos(bx)$, it changes how long it takes for the wave to repeat, which is called the period. The new period becomes $\frac{2\pi}{|b|}$. This can confuse students, especially when they try to understand how often the waves happen.

To make these changes easier to grasp, it's helpful for students to practice using graphs and drawing out the transformations. Using graphing tools can also help them see how each change affects the graph.