Leading Term Analysis
Polynomial end behavior describes how a function's graph behaves as $x$ approaches positive or negative infinity. This is determined entirely by the leading term's coefficient and degree.
The leading term is the term with the highest exponent. For a polynomial $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$, the leading term is $a_n x^n$.
For example, in the polynomial $f(x) = 5x^4 + 12x^2 - 3x$, the term $5x^4$ dominates. As $x$ becomes very large, $x^4$ grows exponentially faster than $x^2$ or $x$, making the other terms negligible.
End Behavior Rules
The behavior is decided by two factors:
- Leading Coefficient ($a$):
- If positive: The right side ($x \to +\infty$) rises.
- If negative: The right side ($x \to +\infty$) falls.
- Degree ($n$):
- If even: Both ends behave the same (both up or both down).
- If odd: Ends behave oppositely (one up, one down).
End Behavior Reference Table
| Degree | Leading Coeff. | Left ($x \to -\infty$) | Right ($x \to +\infty$) | Description |
|---|---|---|---|---|
| Even | Positive (+) | $f(x) \to +\infty$ | $f(x) \to +\infty$ | Both ends rise (Up/Up) |
| Even | Negative (-) | $f(x) \to -\infty$ | $f(x) \to -\infty$ | Both ends fall (Down/Down) |
| Odd | Positive (+) | $f(x) \to -\infty$ | $f(x) \to +\infty$ | Falls left, Rises right |
| Odd | Negative (-) | $f(x) \to +\infty$ | $f(x) \to -\infty$ | Rises left, Falls right |
Practice Examples
Example 1: Negative Even Degree
$$f(x) = -55x^4 - 3x^3 + 2x - 1$$
Leading Term: $-55x^4$
Degree: $4$ (Even) $\to$ Same behavior on both ends.
Coefficient: $-55$ (Negative) $\to$ Right end falls.
Result: Both ends fall.
As $x \to -\infty, f(x) \to -\infty$; As $x \to +\infty,
f(x) \to -\infty$
Example 2: Positive Odd Degree
$$g(x) = x^3 - 2x$$
Leading Term: $1x^3$
Degree: $3$ (Odd) $\to$ Opposite behavior.
Coefficient: $1$ (Positive) $\to$ Right end rises.
Result: Falls left, rises right.
As $x \to -\infty, g(x) \to -\infty$; As $x \to +\infty,
g(x) \to +\infty$
Example 3: Factored Form
$$h(x) = -2x^2(x-1)(x+3)$$
To find the leading term, multiply the highest degree terms of each factor: $(-2x^2) \cdot
(x) \cdot (x) = -2x^4$.
Leading Term: $-2x^4$
Degree: $4$ (Even) $\to$ Same behavior.
Coefficient: $-2$ (Negative) $\to$ Right end falls.
Result: Both ends fall.
As $x \to -\infty, h(x) \to -\infty$; As $x \to +\infty,
h(x) \to -\infty$