Explanation

End behavior describes what happens to a function as x becomes:

• Very large (x → ∞)

• Very negative (x → −∞)

To determine end behavior:

• Compare the degrees of the numerator and denominator

Cases:

1. Degree numerator < degree denominator → y = 0 (horizontal asymptote)

2. Degrees equal → y = ratio of leading coefficients

3. Degree numerator > degree denominator → slant or curved asymptote

Example:
f(x) = (2x² + 1)/(x² − 5)

Degrees equal → y = 2/1 = 2

Quiz

1. What does end behavior describe?

2. What happens when numerator degree < denominator degree?

3. Find the horizontal asymptote of (3x²)/(x² + 1).

4. What happens when numerator degree > denominator degree?

5. Does end behavior describe local behavior?

Answer Key

1. Behavior as x → ±∞

2. y = 0

3. y = 3

4. Slant or curved asymptote

5. No