Polynomial Functions & Their Graphs

The degree and the leading coefficient decide the shape before you plot a single point.

Look at a roller coaster profile and count the hills.

A straight run has no turns. One hill and you have a parabola. Two turns needs a cubic.

a track with three hills needs a big enough rule turn turn turn turn turn five turns — a line has none, a parabola has one

The track drawn here turns five times, so no cubic could hold it — the rule has to be bigger.

A polynomial of degree n can turn at most n - 1 times. Five turns needs degree six or more.

That is the whole idea of this unit: the degree caps how wiggly a graph can be, and the leading coefficient decides where the ends go. Everything else is detail.

What the ends do, by degree and sign two things decide the ends: the degree, and the sign of the lead even, + even, − odd, + odd, − even degree: both ends agree · odd degree: they disagree

What the ends do

Far from the origin the highest power swamps everything else, so the leading term alone decides the ends.

Even degree: both ends point the same way — both up if the lead is positive, both down if negative.

Odd degree: the ends disagree. Positive lead falls on the left and rises on the right.

Two questions, four answers

Is the degree even or odd? Is the lead positive or negative? Those two answers fix both ends, and no other part of the polynomial matters.

Roots are where the graph meets the axis

Setting y = 0 and solving gives the x-intercepts, so a factored polynomial shows its roots directly.

y = (x-1)(x+2)(x-4) crosses at 1, -2 and 4 — no working required.

A degree n polynomial has at most n real roots, which is why a cubic can never cross four times.

Factored form is the useful form

Expanding hides the roots. If a question asks where the graph crosses, factor rather than expand.

Multiplicity: cross, or just touch

A root that appears once makes the graph cross the axis cleanly.

A root appearing twice makes it touch and turn back, like a parabola resting on the axis.

Odd multiplicity crosses; even multiplicity bounces. That is why (x-2)2 sits on the axis at 2 instead of passing through.

Count the factor, not the root

In y = (x-3)2(x+1) the root 3 has multiplicity 2 so the graph bounces there, while -1 is single so it crosses.

Turning points

Between two roots the graph has to turn round to come back, so roots and turns are tied together.

A degree n polynomial has at most n-1 turning points — a cubic turns at most twice.

It can have fewer. y = x3 has none at all, because it rises the whole way.

Working backwards from a picture

Count the turns on a graph, add one, and you have the smallest degree that could draw it.

Sketching without plotting

Find the roots from the factors and mark them. Decide cross or bounce from each multiplicity.

Fix the two ends from the degree and the leading sign, then join it up smoothly.

One extra point, usually the y-intercept at x = 0, pins the vertical scale.

The sketch checks itself

If your curve needs more turns than n-1 to connect the marks, something is wrong — go back and recheck a multiplicity.

Worked examples

End behaviour

Describe the end behaviour of y = -2x4 + 7x3 - 5.

  1. The degree is 4, which is even. Even degree means both ends agree.
  2. The leading coefficient -2 is negative. Negative flips both ends downward.
  3. So the graph falls at both ends. The 7x3 shapes the middle but has no say about the ends.

Down on the left, down on the right

Reading a factored polynomial

For y = (x+1)(x-2)2, give the roots and say whether the graph crosses or touches at each.

  1. x + 1 = 0 gives x = -1, appearing once. Multiplicity 1, so it crosses.
  2. (x-2)2 = 0 gives x = 2, appearing twice. Multiplicity 2, so it touches and turns back.
  3. Degree is 1 + 2 = 3, positive lead, so it rises to the right. Odd degree means the ends disagree.

Crosses at -1, touches at 2

Smallest possible degree

A graph crosses the x-axis three times and has four turning points. What is the smallest degree it could have?

  1. Four turns needs degree at least 4 + 1 = 5. A degree n polynomial turns at most n-1 times.
  2. Three crossings needs degree at least 3, which is already satisfied. The turning points are the stronger constraint here.
  3. The ends disagree — three crossings with four turns needs an odd degree. Degree 5 meets every requirement.

5

Words to know

Degree
The highest power of x. It caps the number of roots and the number of turns.
Leading coefficient
The number in front of the highest power. Its sign decides which way the right end goes.
End behaviour
What the graph does far out to the left and right.
Multiplicity
How many times a root repeats. It decides whether the graph crosses or just touches.
Turning point
Where the graph changes from rising to falling or back. At most n - 1 of them.

Mistakes to watch for

Judging the ends from the wrong term. In y = -2x4 + 7x3 the 7x3 looks cheerful, but at x = 100 the x4 term is fourteen times larger. Only the leading term decides the ends.

Assuming a double root crosses. y = (x-2)2 touches the axis at 2 and turns back. Even multiplicity bounces, odd multiplicity crosses — the graph never passes through a squared factor.

Counting turning points as equal to the degree. Degree n allows at most n-1 turns, so a cubic turns twice at most. Reading four turns and calling it degree four is one short.

Expanding before reading the roots. (x-1)(x+2)(x-4) shows its intercepts at a glance; multiplied out it is x3 - 3x2 - 6x + 8 and they are hidden. Factored form is the useful form.

Forgetting that fewer turns is allowed. y = x3 has no turning points at all, so "degree 3" does not promise two turns — it only caps them.

The word comes from Greek poly and Latin nomen — "many names", meaning many terms.

Ships’ hulls, car bodies and font outlines are all drawn with polynomial curves, because they are smooth and a computer can evaluate them fast.

The fundamental theorem of algebra says a degree n polynomial has exactly n roots, if you count repeats and allow complex ones. Gauss proved it at twenty-two.

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