Understanding Vertical and Horizontal Shifts in Trigonometric Graphs

When we look at trigonometric graphs, like the sine and cosine graphs, we can change where they sit on the coordinate plane. Two main ways to do this are vertical and horizontal shifts.

Vertical Shifts

• What It Is: This involves adding or subtracting a number, which we’ll call $k$, to the function.
• What Happens: If you add a positive number ($k > 0$), the graph moves up. If you subtract a number ($k < 0$), it moves down. The shape of the graph doesn’t change.
• Example: If we have the function $f(x) = \sin(x) + 2$, this means the sine graph goes up by 2 units.

Horizontal Shifts

• What It Is: This means adding or subtracting a number, called $d$, inside the function.
• What Happens: If you subtract a positive number ($d > 0$), the graph moves left. If you add a number ($d < 0$), it moves right.
• Example: For the function $f(x) = \cos(x - \frac{\pi}{2})$, the cosine graph shifts to the right by $\frac{\pi}{2}$ units.

Both types of shifts keep the important features of the graphs, like how high or low they go (amplitude) and how long it takes to complete one wave (period).