Understanding End Behavior of Functions

End behavior is all about how a function behaves when the input values get really big or really small.

Knowing about end behavior is important because it helps us see the limits and main features of a function.

Key Features of Graphs

1. Intercepts:

   • X-intercepts are points where the graph crosses the x-axis. This is where the output $f(x) = 0$.
   • Y-intercepts are points where the graph crosses the y-axis. This happens when $x = 0$.

2. Asymptotes:

   • Vertical Asymptotes show where the function's output gets really big (or close to infinity), such as when $x = c$.
   • Horizontal Asymptotes show the value the function gets closer to as $x$ goes towards really big positive or negative numbers. For example, in the function $f(x) = \frac{1}{x}$, the horizontal asymptote is $y = 0$.

Analyzing End Behavior

Looking at the end behavior of a function can help us understand:

• Limitations: Knowing what happens as $x$ becomes really big or really small helps us figure out the highest or lowest values a function can reach. For example, polynomial functions like $f(x) = x^2$ go to infinity as $x$ goes to either positive or negative infinity.

• Graph Shape: We can group functions based on their end behavior:

  • Linear Functions: For example, $f(x) = mx + b$, which keep increasing or decreasing without stopping.
  • Rational Functions: For example, $f(x) = \frac{1}{x^2}$ approaches the horizontal line at $y = 0$.

Seeing these patterns helps us predict how functions will behave in different situations, which is very useful in math.