End behavior describes what a function does at the far left and the far right of its graph: what the outputs do when the inputs get very large in magnitude. We start with the right end.
As the input values of a nonconstant polynomial function increase without bound, the output values will either increase without bound or decrease without bound. There is no third option: a nonconstant polynomial never levels off at a number.
Read the first one as "as x increases without bound, p(x) increases without bound." Read the second as "as x increases without bound, p(x) decreases without bound." The symbol ∞ is not a number. It just says "without bound."
Pick a polynomial and look at the table: the inputs move to the right without bound. Then zoom the graph out and watch the right end settle into a direction.
| x | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|
| p(x) | 960 | 999,600 | 999,996,000 | 1.00×10¹² |
Now go the other way. As the input values of a nonconstant polynomial function decrease without bound (x → −∞, moving left along the axis), the output values will again either increase without bound or decrease without bound.
The two ends are separate questions. For x³ − 4x the left end falls and the right end rises. For x⁴ − 100x² both ends rise. The next two sections explain why.
Pick a polynomial and look at the table: the inputs move to the left without bound. Then zoom the graph out and watch the left end settle into a direction.
| x | −10 | −100 | −1,000 | −10,000 |
|---|---|---|---|---|
| p(x) | −960 | −999,600 | −999,996,000 | −1.00×10¹² |
Why does a polynomial have just two possible directions at each end? Because for large inputs the leading term dominates. As the input values increase or decrease without bound, the values of the leading term become far larger in size than the values of all the lower-degree terms put together. So the polynomial acts like its leading term.
Take x³ − 50x² + 7. At x = 10 the lower terms win: 1,000 − 5,000 + 7 is negative. But a cube grows faster than a square, so by x = 100 and x = 1,000 the x³ term has taken over.
Slide the input x = 10k far out. Compare the leading term with everything else added together. The ratio p(x) ÷ (leading term) heads toward 1 as the inputs get large.
| piece | value at x = 100 |
|---|---|
| leading term x³ | 1,000,000 |
| all lower-degree terms | −499,993 |
| p(x) | 500,007 |
| p(x) ÷ leading term | 0.5 |
The degree and sign of the leading term determine the end behavior. Only two facts about the leading term are needed: is the degree even or odd, and is the coefficient positive or negative.
| Leading term | Left end (x → −∞) | Right end (x → ∞) |
|---|---|---|
| even degree, positive | ∞ | ∞ |
| even degree, negative | −∞ | −∞ |
| odd degree, positive | −∞ | ∞ |
| odd degree, negative | ∞ | −∞ |
Why the pattern? The right end follows the sign of the leading coefficient. On the left, x is negative. An even power turns it positive, so the left end matches the right. An odd power keeps it negative, so the left end is the opposite of the right. That is the same even-and-odd idea from Lesson 1.5.
Choose the degree and the sign of the leading term. Turn the lower-degree terms on and off: they change the wiggles in the middle, but the two ends do not change.
You do not have to multiply out a factored polynomial to find its ends. Multiply only the leading terms of the factors. For p(x) = −2(x + 1)²(x − 3)³ that is −2 · x² · x³ = −2x⁵. The degree is 2 + 3 = 5 (odd) and the leading coefficient is −2 (negative). So the left end rises and the right end falls.
p(x) = a(x + 1)m₁(x − 2)m₂. You never have to multiply it out: the leading term is a times x to the degree m₁ + m₂.
Going backward works too. From a graph you can read the end behavior, and from the end behavior you can say whether the degree is even or odd and whether the leading coefficient is positive or negative.
| You see | It means |
|---|---|
| Both ends go the same way | Even degree |
| The ends go opposite ways | Odd degree |
| Right end goes up | Positive leading coefficient |
| Right end goes down | Negative leading coefficient |
| Ends opposite | At least one real zero: the graph must cross the x-axis |
Look only at the two ends of each graph and predict each row. Click a row to check yourself. Row four brings back Lesson 1.5: a crossing counts for at least one zero and a tangent counts for at least two.