Understanding how linear functions change can really help you see how they work. When I was in 9th grade, I enjoyed trying out different changes with graphs. It’s all about how the equation changes the graph. Let’s make it simple:

How Linear Functions Change

1. Shifts:

   • Up and Down Shifts: When you add or take away a number from the function, it moves up or down. For example, if you start with $f(x) = x$ and change it to $f(x) = x + 3$, the whole graph moves up by 3 units. If you change it to $f(x) = x - 2$, the graph moves down by 2 units.

   • Side Shifts: These happen when you add or subtract a number inside the function. For instance, $f(x) = x - 4$ moves the graph to the right by 4 units, while $f(x) = x + 1$ moves it to the left by 1 unit.

2. Stretches and Squeezes:

   • Up and Down Stretch/Squeeze: This occurs when you multiply the function by a number. If you multiply by a number bigger than 1, like in $f(x) = 2x$, the graph stretches up, making it rise faster. If you multiply by a small number, like $f(x) = 0.5x$, it squeezes up, making it rise slower.

   • Side Stretch/Squeeze: This happens inside the function. So, $f(x) = x/3$ stretches the graph sideways, while $f(x) = 3x$ squeezes it sideways.

Using Graphs to See Changes

The best way to really understand these changes is to use graphing tools. You can try a graphing calculator or an online tool like Desmos. Here’s how I did it:

• Start with the Basic Function: Usually, we use $f(x) = x$ as our starting point. Graph that first.
• Make One Change at a Time: Shift it up, down, left, or right one at a time. Watch how the graph changes.
• Try Stretches and Squeezes: Change the numbers and see how it changes the slope of the graph. It’s like playing around with it!

Important Points

• Keep Practicing: The more you play with different changes, the better you will understand them. It’s also helpful to draw the changes by hand so you can see the differences.
• Look for Patterns: Pay attention to how the changes you make to the function affect the graph’s shape and position. This will help you predict what will happen next time you change something.

In the end, seeing how linear functions change not only makes Algebra fun but also helps you understand functions better. Give it a try! Grab some graph paper or use an online tool and start changing those lines! It’s a lot of fun!