Understanding Different Types of Functions

There are three main types of functions: linear, quadratic, and exponential. Each type changes at different rates. Let’s break them down.

1. Linear Functions:

   • The formula looks like this: (y = mx + b).
   • Here, (m) is called the slope.
   • In linear functions, the rate of change stays the same.
   • For example, if the slope (m) is 2, every time (x) goes up by 1, (y) also goes up by 2.

2. Quadratic Functions:

   • The formula for a quadratic function is: (y = ax^2 + bx + c).
   • These functions don’t change at a constant rate.
   • Instead, the rate changes based on the value of (x).
   • For instance, if (a = 1), at (x = 0), the change is just (b).
   • But when (x) increases, the change will get bigger. At (x = 2), the change becomes higher because of the (x^2) part.

3. Exponential Functions:

   • The formula looks like this: (y = ab^x) where (b) is greater than 1.
   • These functions increase quickly.
   • The way they change is much faster than linear and quadratic functions.
   • For example, if (a = 1) and (b = 2), at (x = 1), the rate of change is about 0.693.

In summary:

• Linear functions change at a steady pace.
• Quadratic functions change their pace.
• Exponential functions pick up speed as they grow.

As (x) gets bigger, exponential functions outgrow both linear and quadratic functions very quickly.