Unit 1 · Topic 1.6
Polynomial Functions and End Behavior
Use a polynomial's degree and leading coefficient to predict what happens to its outputs as inputs increase or decrease without bound.
1. What End Behavior Describes
End behavior describes the outputs of a function far to the left and far to the right of its graph.
- Right end: inputs increase without bound, written \(x\to\infty\).
- Left end: inputs decrease without bound, written \(x\to-\infty\).
Both ends must be stated. Describing only the right tail is an incomplete end-behavior statement.
2. Read Limit Notation Correctly
\[\lim_{x\to\infty}P(x)=\infty\]
This says that \(P(x)\) increases without bound as \(x\) increases without bound. It does not mean that infinity is an input or an output value.
\[\lim_{x\to-\infty}P(x)=-\infty\]
This says that outputs decrease without bound as inputs move farther left.
3. Why the Leading Term Controls the Tails
Write a nonconstant polynomial in descending powers:
\[P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\qquad a_n\ne0.\]
For inputs of very large magnitude, \(x^n\) grows in magnitude faster than every lower power. The leading term \(a_nx^n\) therefore dominates the sum.
\[P(x)\sim a_nx^n\qquad\text{as }x\to\pm\infty.\]
Lower-degree terms can create intercepts, extrema, and unusual behavior near the middle, but they cannot change the eventual direction of either tail.
4. The Four End-Behavior Patterns
| Degree | Leading coefficient | As \(x\to-\infty\) | As \(x\to\infty\) | Tail pattern |
|---|
| Even | Positive | \(P(x)\to\infty\) | \(P(x)\to\infty\) | Left up, right up |
| Even | Negative | \(P(x)\to-\infty\) | \(P(x)\to-\infty\) | Left down, right down |
| Odd | Positive | \(P(x)\to-\infty\) | \(P(x)\to\infty\) | Left down, right up |
| Odd | Negative | \(P(x)\to\infty\) | \(P(x)\to-\infty\) | Left up, right down |
Memory structure: even-degree tails point in the same vertical direction; odd-degree tails point in opposite directions. The leading coefficient determines the right tail.
5. A Reliable Decision Process
- Identify the leading term.
- Decide whether its degree is even or odd.
- Determine whether its coefficient is positive or negative.
- Use parity to decide whether the tails match or oppose.
- Use the coefficient sign to place the right tail.
- Write two complete limit statements.
6. Worked Example in Standard Form
Consider \(P(x)=-3x^5+4x^3-7x+2\).
- Degree: 5, which is odd.
- Leading term: \(-3x^5\).
- Leading coefficient: \(-3\), which is negative.
\[\lim_{x\to-\infty}P(x)=\infty,\qquad \lim_{x\to\infty}P(x)=-\infty.\]
The left tail rises and the right tail falls. The terms \(4x^3-7x+2\) affect the middle but not the final tail directions.
7. Find the Leading Term from Factored Form
Expansion is usually unnecessary. Multiply only the leading part of each factor.
\[F(x)=-2(x-1)^3(3x+5)^2.\]
\[-2(x)^3(3x)^2=-2(x^3)(9x^2)=-18x^5.\]
Thus \(F\) has odd degree 5 and a negative leading coefficient. Its left tail rises and its right tail falls.
| Expression feature | How to use it |
|---|
| Constant multiplier | Include its sign and value in the leading coefficient. |
| Factor multiplicity | Multiply the factor's degree by its exponent. |
| Several factors | Add their degrees and multiply their leading coefficients. |
| Added constant outside a product | It is lower degree and does not affect end behavior. |
8. Infer Information from a Graph
The two tails reveal two facts, but usually not the exact degree.
- Same-direction tails imply an even degree.
- Opposite-direction tails imply an odd degree.
- A rising right tail implies a positive leading coefficient.
- A falling right tail implies a negative leading coefficient.
For example, a graph with both tails down has an even degree and a negative leading coefficient. It could have degree 2, 4, 6, or another positive even degree.
9. End Behavior Does Not Describe the Entire Graph
Two polynomials can share the same end behavior while having different zeros and turning behavior.
\[A(x)=x^4,\qquad B(x)=x^4-20x^2+64.\]
Both have leading term \(x^4\), so both tails rise. However, their middle sections are very different. End behavior is a statement about sufficiently large positive and negative inputs, not every input.
10. A Narrow Graphing Window Can Mislead
Consider \(G(x)=x^4-100x^3\). Its leading term is \(x^4\), so both tails eventually rise, but the right side remains negative for many positive inputs.
| \(x\) | \(-200\) | \(-10\) | \(10\) | \(50\) | \(200\) |
|---|
| \(G(x)\) | \(2{,}400{,}000{,}000\) | \(110{,}000\) | \(-90{,}000\) | \(-6{,}250{,}000\) | \(800{,}000{,}000\) |
|---|
A window ending at \(x=50\) could make the right tail appear to fall. Expanding the window and checking the leading term prevents this error.
11. Transformations and Tail Behavior
| Add a constant | Vertical shifts do not change either infinite tail direction. |
|---|
| Replace \(x\) with \(x-h\) | Horizontal shifts preserve degree and leading coefficient. |
|---|
| Multiply by a positive constant | Tail directions remain the same. |
|---|
| Multiply by a negative constant | Both tail directions reverse. |
|---|
Always recompute the leading term when transformations are combined rather than relying only on a memorized picture.
12. Technology and Representation Workflow
- Use the formula to determine the theoretical end behavior first.
- Graph the function and inspect both tails.
- Increase the horizontal and vertical window if the visible graph conflicts with the leading-term prediction.
- Use a table with inputs of increasing magnitude to gather numerical evidence.
- Report the result verbally and with two limit statements.
Technology confirms and illustrates the analysis; the leading term supplies the general justification.
13. Common Errors and AP Reasoning
- Using the constant term or the first printed term instead of the highest-degree term.
- Ignoring a negative multiplier outside factored form.
- Confusing an even function with an even-degree polynomial.
- Writing \(x=\infty\) as though infinity were a number.
- Reversing \(x\to-\infty\) and \(P(x)\to-\infty\).
- Describing only one tail.
- Trusting a narrow graphing window over the leading term.
Complete response pattern: “The leading term is \(a_nx^n\). Since \(n\) is even/odd and \(a_n\) is positive/negative, as \(x\to-\infty\), ... , and as \(x\to\infty\), ... .”