When we talk about horizontal changes in functions, we are looking at how the graph moves along the x-axis. These changes can include moving left or right, or making the graph wider or narrower.

Moving the Graph Left or Right

1. Moving Right: If we have a function, like $f(x)$, and we change it to $f(x - h)$ (where $h$ is a positive number), the graph moves to the right by $h$ units.

   • Example: If $f(x) = x^2$, then when we write $f(x - 2) = (x - 2)^2$, the graph shifts right by 2 units.

2. Moving Left: If we change the function to $f(x + h)$ (again, where $h$ is positive), the graph moves to the left by $h$ units.

   • Example: For $f(x) = x^2$, if we write $f(x + 2) = (x + 2)^2$, the graph shifts left by 2 units.

Stretching and Squishing the Graph

Horizontal changes can also mean making the graph wider or narrower:

• Widening the Graph: When you change $f(x)$ to $f(kx)$ (where $0 < k < 1$), the graph gets wider, moving away from the y-axis.

  • Example: If $f(x) = x^2$ and $k = \frac{1}{2}$, then $f(\frac{1}{2}x) = (\frac{1}{2}x)^2 = \frac{1}{4}x^2$ makes the graph wider.

• Narrowing the Graph: When you change the function to $f(kx)$ (where $k > 1$), the graph gets narrower, moving closer to the y-axis.

  • Example: For $f(x) = x^2$ and $k = 2$, then $f(2x) = (2x)^2 = 4x^2$ squishes the graph.

In Summary

In short, horizontal changes can really change what a function looks like and where it is located. Knowing how to move and stretch graphs can make it easier to work with them. Keep practicing with different functions to see how these changes affect the graphs!