Topic 1.6: Polynomial Functions and End Behavior

Sixty feet tall

This is the countryside just outside Florence, a little after sunrise, and the figure standing by that stone wall is made of glass and exactly as tall as you are. Let’s start making him bigger, first twice as tall as you, then four times, until his head reaches the roof of that farmhouse. At eight times your height he’s taller than the farmhouse. And if you look inside his legs, you can see his bones starting to glow.

At ten times your height he’s sixty feet tall, eye to eye with the cypress trees, and now he’s going to try to take one step forward. In sixteen thirty-eight, Galileo, an old man under house arrest in a villa on these hills, wrote that this has to happen, that if you make a giant tall enough, he will fall and be crushed under his own weight. And every movie monster the size of a building? Exactly the same problem.

By the end of this lesson, you’ll be able to prove, with one line of algebra, that a sixty-foot giant breaks his legs on his very first step. And you’ll know exactly what you’d have to change to build one who could stand. It all comes down to what a polynomial does when its input gets really big. So let’s put him back together and find out exactly when he gets into trouble.

Spare strength

Here he is again. And this time, let’s look inside him. When he’s twice as tall, he’s also twice as wide and twice as thick, so he’s made of two times two times two, eight times as much glass, and he weighs eight times as much. A bone’s strength comes from its cross-section, the area you’d see if you sliced straight through it, and that only grows two times two, four times, so his bones fall behind his weight.

Almost three hundred years later, the biologist J. B. S. Haldane put a number on it, writing that the human thigh-bone breaks under about ten times the human weight. So your bones could hold ten of you.

Let’s call the scale factor k. The bones grow with their cross-section, so instead of ten body weights, they can hold ten times k squared, while the giant himself weighs k cubed, k times k times k. What’s left over is his spare strength, S of k equals ten k squared minus k cubed. Real bones aren’t perfect scaled copies, but this model gets the main idea right.

At your own size the bones hold ten, and you weigh one. At twice your height they can hold forty, and he weighs only eight. And at six times it’s three hundred sixty against two hundred sixteen, so the bones stay well ahead of him. Subtract his weight from what the bones can hold, forty minus eight is thirty-two, and his spare strength climbs from nine all the way to a hundred forty-four, and it looks like it’ll just keep on climbing.

If this graph were all you had, you’d say the bigger he gets, the stronger he gets. That’s the exact opposite of what we just watched happen. So let’s ask the giant himself, with his spare strength drawn in the sky above him.

Take the dial and grow him slowly. See if you can find the size where he has the most strength to spare. Now keep growing him. The spare strength shrinks faster and faster, until at exactly ten times your height it hits zero, and with nothing to spare, his first step breaks him.

Through a wider window, the graph climbs, turns over near six point seven, crosses zero at ten, and by twenty times your height he’s four thousand body weights short. It never comes back up, so we say that as k increases without bound, S of k decreases without bound. That first graph only ever showed us a small window, and the formula knows where the ends go even when a window can’t show them.

Why the cube wins

In the formula, the bones’ term starts with a ten-to-one head start, and at five it’s still ahead, two fifty to one twenty-five. At ten they tie at a thousand each, exactly when his legs break. By a hundred it’s a hundred thousand against a million, and by a thousand it’s ten million against a billion, so k cubed runs away with it.

There’s a way to see all of this in one line. If I factor negative k cubed out of both terms, I get negative k cubed times one minus ten over k. If you multiply it back out to check, negative k cubed times negative ten over k gives plus ten k squared, the bones’ term we started with.

Now watch the part in parentheses as k grows. At five, ten over k is two, and one minus two is negative one. At ten, ten over k is one, so it’s zero. By a thousand, ten over k is just a hundredth, which leaves zero point nine nine, closing in on one.

So for a big giant, S of k is negative k cubed times almost exactly one, and the term with the highest power, the leading term, takes over completely. And that one line is the proof I promised you. At ten the parentheses are zero, so he has nothing to spare, and for any bigger giant they’re positive, and negative k cubed times a positive number is negative, so he can’t even stand.

Here’s one for you. How many times as tall would the giant have to be before the bones’ term, ten k squared, is just one percent of the weight term, k cubed?

None of this is special to giants. Here’s a polynomial with a lot more going on, x to the fifth minus five x cubed plus four x, with five zeros and four turns. Before I zoom out, pick the one term you think this graph will look like from far away. Keep that term in your head.

I’ll zoom out ten times sideways and a hundred thousand times up and down, the zoom that keeps x to the fifth looking exactly the same, because ten to the fifth is a hundred thousand. Negative five x cubed flattens onto the axis, four x disappears completely, and the whole polynomial melts onto x to the fifth. If that’s the term you picked, you just read the end behavior straight off the formula.

Every polynomial follows the rule that broke our giant. Far enough out in either direction, the leading term outgrows all the others put together, so its degree and its sign decide where the ends go.

Where the ends go

So how far out is far enough? Let’s build this exact graph at full size, one meter for every unit, on the salt flat of Uyuni in Bolivia, the biggest mirror on Earth. All five zeros fit within a few steps. And that hump is about twice as tall as the person beside it.

Now let’s ride the right arm. Just past six meters out, it’s as high as Mount Everest, just past ten meters it reaches the edge of space, and a little past thirteen it passes the space station. And only fifty-two meters from where we started, about the length of an Olympic pool, this graph reaches the Moon.

Up here, it’s practically the same line as x to the fifth. That’s its leading term. The left arm goes the other way. Twenty-three meters to the left of where we started, it reaches all the way down to the center of the Earth. Neither arm ever levels off or turns back, and that’s what it means for the outputs to increase or decrease without bound.

On the board we write the right arm in words first. As x increases without bound, p of x increases without bound. And in limit notation, we write the very same sentence like this. The x arrow infinity under the limit says the input increases without bound, and equals infinity says p of x increases without bound.

The left arm is the same idea in the other direction. As x decreases without bound, p of x decreases without bound, and the notation gets two minus signs. Think of that infinity as a direction the arm keeps heading in forever. A polynomial that isn’t just a constant never levels off at either end, though you’ll meet functions that do in the next lesson.

Four ways out

From up here, you can see both arms. The left one dives into the Earth. The right one climbs into the sky. Only two things decide those arms, the degree and the sign of the leading coefficient, and I’d like you to find the rule yourself before I tell you.

Use those two controls and try to make both arms dive into the Earth. Now I’ll change everything except the leading term. The hump and the ditch twist into new shapes, and the arms out at the ends don’t move at all.

Here’s the rule, and the trick is to read from the right. The right end follows the sign of the leading coefficient, up when it’s positive and down when it’s negative. Then the left end copies the right end when the degree is even. When the degree is odd, it flips it.

Out on the left x is negative, so the reason is what powers do to negative numbers. Negative ten squared is a positive hundred, because an even power forgets the sign, but negative ten cubed is negative one thousand, because an odd power keeps it. You can even do all four with your arms.

Both up for x squared. Both down for negative x squared. Left down and right up for x cubed. And the reverse for negative x cubed. Try it once, and you’ll never forget it.

Find the leading term

Now let’s find the leading term in a formula that’s trying to fool you, f of x equals five plus four x cubed minus x to the fourth. It isn’t the five, even though it’s written first, and it isn’t four x cubed, even though four is the biggest number in front of an x. The leading term is the one with the highest power, negative x to the fourth.

Written from the highest power down, it starts with negative x to the fourth. Degree four is even, so the ends agree, and the coefficient is negative, so both ends head down without bound.

Factored form hides the leading term even better. Here’s g of x equals negative two, times x minus one, squared, times three minus x, times x plus two, squared. You never have to multiply that out, because the leading term is just the leading pieces of the factors, multiplied together. From x minus one, squared, for example, the leading piece is just x squared.

Let’s do one we already know. The giant’s formula factors as S of k equals k squared times ten minus k, and the leading piece of ten minus k is negative k, because the k is the part that grows. Multiply k squared by negative k and you get negative k cubed, exactly the term that broke him.

Work out the leading term in your head first. Then decide what the two ends of g do. Now try one on your own. What’s the leading term of q of x equals two x minus one, cubed, times four minus x?

The formula that buried a rocket

Remember the rocket from lesson one point two? The altitude curve we drew for its climb came from this degree-five polynomial, and it matched the real flight beautifully for two and a half minutes. Look at the last coefficient. It’s less than one billionth, and negative, the tiniest number in the whole formula, and it sits on t to the fifth.

This is exactly how the AP exam likes to ask it. If you trusted this formula forever, what would it predict for the rocket’s altitude as t increases without bound?

Let’s ask it. For the first hundred fifty-eight seconds the formula and the real rocket fly together, and then they part ways. The formula has the rocket top out at about a hundred ninety-five kilometers, four minutes after launch, and then fall back into the Atlantic at five minutes and eleven seconds, while the real rocket keeps going all the way to orbit. That tiny last term grows the fastest, and a little after six minutes it’s the biggest term in the formula, and it never gives the lead back.

On the exam, they’ll often ask for the end behavior in limit notation. And the best answers back it up with the reason, which is that the leading term has an odd degree and a negative coefficient. Then say what the formula is good for, the first hundred fifty-eight seconds of this flight, because outside that window it isn’t telling you anything real.

A giant who stands

Now for the second half of the promise, a giant who can stand. Galileo had two ideas, and the first was the sea. He pointed out that the biggest animals on Earth live in water. The water carries most of their weight for them.

Say the water carries ninety-five percent of our giant’s weight, so the weight term shrinks to zero point zero five k cubed. At a hundred times your height, he’s still standing. But the leading term is still negative, just smaller, so at two hundred times your height the cube catches him anyway. A smaller leading coefficient only buys you time.

Galileo’s other idea was to thicken the bones as the giant grows, enough that their strength keeps up with his weight, so the bones’ term becomes ten k cubed and the spare strength is nine k cubed. The catch is how he looks, and Galileo saw it coming, writing that the shape keeps changing “until the form and appearance of the animals suggest a monstrosity.” Now he’s your giant. Try the fixes and find out which one gets him standing at three hundred times your height.

And this time, when he takes his first step, it holds. So the whole lesson fits on one giant. First you find the leading term, and you read the right end from its sign. Then an even degree copies that end to the left, and an odd degree flips it. A polynomial that isn’t constant never levels off, but in the next lesson you’ll meet functions that settle down, and you’ll see exactly how they manage it.