When looking at how graphs change, vertical and horizontal shifts are important for moving a graph around on a coordinate plane. Let’s simplify how these shifts work.

Vertical Shifts

A vertical shift moves a graph up or down.

This happens when you add or subtract a number from the entire function.

For example, if you have a function $f(x)$ and you add a number $k$, it looks like this:

$g(x) = f(x) + k$

For example:

• Imagine $f(x) = x^2$. If we make a new function $g(x) = x^2 + 3$, the graph of $g(x)$ will move up by 3 units.
• On the other hand, if we take $f(x) = x^2$ and subtract a number, like $g(x) = x^2 - 2$, then the graph moves down by 2 units.

Horizontal Shifts

Horizontal shifts move the graph left or right.

This type of shift can be a little tricky because it involves changing the input $x$ in the function.

Here’s how it looks:

$g(x) = f(x - h)$

Here’s how it works:

• If $h$ is a positive number, the graph shifts to the right. For instance, if we have $f(x) = x^2$ and use $g(x) = (x - 2)^2$, the graph moves right by 2 units.
• If $h$ is a negative number, the graph shifts to the left. So, if we use $g(x) = (x + 3)^2$, the graph shifts left by 3 units.

Summary

To sum it all up, here’s the main difference between the shifts:

• Vertical shifts move the graph up or down by changing the output of the function.
• Horizontal shifts move the graph left or right by changing the input of the function.

Seeing these changes visually on a graph can really help you understand what’s happening. Remember: adding or subtracting Outside the function moves it vertically, while adding or subtracting Inside the function shifts it horizontally. Enjoy graphing!