To understand how we can change the graphs of trigonometric functions, let’s break this down into two main ways: translation and reflection.

Translation

1. Horizontal Translation:

   • When we have a function called $f(x)$, using $f(x - c)$ shifts the graph to the right by $c$ units.
   • On the other hand, $f(x + c)$ moves it to the left by $c$ units.

2. Vertical Translation:

   • If we add a number $k$ to the function, so it looks like $f(x) + k$, the graph moves up by $k$ units.
   • Conversely, $f(x) - k$ will move it down by $k$ units.

Example: For the sine function, $y = \sin(x)$, if we translate it up by 2 units, we get $y = \sin(x) + 2$. This moves the middle line of the graph from $y = 0$ to $y = 2$.

Reflection

1. Reflection in the x-axis:

   • When we use $-f(x)$, it flips the graph over the x-axis. For example, $y = -\sin(x)$ turns the peaks (high points) into troughs (low points) and vice versa.

2. Reflection in the y-axis:

   • The function $f(-x)$ flips the graph over the y-axis. This results in odd functions, such as $y = \sin(-x)$, which is the same as $y = -\sin(x)$.

Summary

These changes keep trigonometric functions, like sine and cosine, repeating in a regular way, with a cycle of $2\pi$. The height of the waves (amplitude) and how often they repeat (frequency) can change based on stretches or compressions in any direction.