When we look at linear and non-linear functions, it’s really cool to see how they act in different ways. Here are some important points that make them stand out:

1. Graph Shape:

   • Linear functions are shown as straight lines. A common example is $y = mx + b$. Here, $m$ tells you how steep the line is, and $b$ shows you where it hits the y-axis. It’s really easy to guess what the result will be.
   • Non-linear functions, on the other hand, can curve, bounce, or wiggle around. A well-known example is the quadratic function $y = ax^2 + bx + c$, which makes a U-shaped graph called a parabola.

2. Rate of Change:

   • In linear functions, the change is steady. If you increase $x$ by 1, $y$ will change by the same amount every time.
   • For non-linear functions, the change can be different. For instance, in a quadratic function, as you increase $x$, the change in $y$ can get faster or slower, depending on where you are on the graph.

3. Form of the Equation:

   • Linear equations usually look like $y = mx + b$. There are no exponents higher than 1 for $x$.
   • Non-linear functions can have higher powers, square roots, or even increase really quickly, like $y = a \cdot b^x$ in exponential functions.

4. Intercepts:

   • A linear function will always cross the y-axis at one point. However, non-linear functions might touch the y-axis two times, one time, or not at all, depending on their shape.

Understanding these differences can really help you work with different types of functions!