Learning about vertical shifts in trigonometric graphs is both fun and important! It helps us understand how these kinds of functions work. Let’s explore!

What is a Vertical Shift?

A vertical shift is when the whole graph of a trigonometric function moves up or down on the graph. This happens when we add or take away a number from the function.

The General Form

We can usually write trigonometric functions like this:

• For sine: $y = A \sin(B(x - C)) + D$
• For cosine: $y = A \cos(B(x - C)) + D$

In these equations, the part that matters for vertical shifts is $D$!

Identifying the Shift

• Upward Shift: If $D$ is a positive number (like $D = 3$), the graph moves up by that amount. Cool, right?
• Downward Shift: If $D$ is a negative number (like $D = -2$), the graph goes down by that amount. How neat!

Example

Let’s look at an example:

1. For the function: $y = \sin(x) + 4$

   • Here, since $D = 4$, the whole sine graph moves up 4 units.

2. For another example: $y = \cos(x) - 2$

   • In this case, it means the cosine graph shifts down 2 units.

Recap

1. Check the constant $D$ in the function.
2. A positive $D$ means the graph goes up, and a negative $D$ means it goes down.

Isn't it amazing how small changes can make a big difference in our graphs? Keep practicing, and soon you’ll be great at spotting those vertical shifts!