Polynomial Functions & Their Graphs — Print

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

What the practice questions cover

These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
Factor — A number that divides evenly into another.
y-intercept — The output when x = 0 — where the line crosses the vertical axis.
Polynomial — A sum of terms, each a number times a whole-number power of the variable.
Roots — The x values where y = 0.
Leading coefficient — The number in front of the highest power. Its sign decides which way the right end goes.
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.

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.

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

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.

The degree is 4, which is even. Even degree means both ends agree.
The leading coefficient -2 is negative. Negative flips both ends downward.
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.

x + 1 = 0 gives x = -1, appearing once. Multiplicity 1, so it crosses.
(x-2)² = 0 gives x = 2, appearing twice. Multiplicity 2, so it touches and turns back.
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?

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

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.

Problem 1

What is the degree of 4x³ - 2x⁵ + 7?

Worked answer

The powers are 3, 5 and 0

The highest is 5

Problem 2

Describe the ends of y = x⁴ - 3x.

Both up
Both down
Down left, up right
Up left, down right

Worked answer

Both up

Degree 4 is even, so the ends agree

The lead +1 sends them both up

Problem 3

Describe the ends of y = -x³ + 2.

Up left, down right
Down left, up right
Both up
Both down

Worked answer

Up left, down right

Odd degree means the ends disagree

Negative lead flips the usual pattern

Problem 4

Give the roots of y=(x-2)(x+5), smallest first.

Worked answer

-5,2

x-2=0 gives 2

x+5=0 gives -5

Problem 5

Give the roots of y=x(x-4)(x+1), smallest first.

Worked answer

-1,0,4

x=0, x=4, x=-1

Problem 6

At x=3, what does the graph of y=(x-3)²(x+1) do?

Touches and turns back
Crosses straight through
Has a hole
Is undefined

Worked answer

Touches and turns back

(x-3) appears twice, so multiplicity 2

Even multiplicity bounces off the axis

Problem 7

At x=-1, what does y=(x-3)²(x+1) do?

Crosses straight through
Touches and turns back
Has a hole
Flattens out

Worked answer

Crosses straight through

Multiplicity 1 is odd

Odd multiplicity crosses

Problem 8

What is the greatest number of turning points a degree 6 polynomial can have?

Worked answer

At most n-1 = 5

Problem 9

A graph has 5 turning points. What is the smallest degree it could be?

Worked answer

n-1 ≥ 5

So n ≥ 6

Problem 10

Give the degree of y=(x-1)²(x+3)³.

Worked answer

2+3=5

Problem 11

Find the y-intercept of y=(x-2)(x+3)(x-1).

Worked answer

(-2)(3)(-1)=6

Problem 12

How many real roots can a degree 4 polynomial have at most?

Worked answer

Degree n gives at most n real roots

Problem 13

Write a polynomial of least degree with roots -2, 1 and 4, leading coefficient 1. Give it factored.

Worked answer

(x+2)(x-1)(x-4)

Root -2 gives (x+2)

Root 1 gives (x-1)

Root 4 gives (x-4)

Problem 14

Write the factored polynomial of least degree that touches at x=2 and crosses at x=-3, leading coefficient 1.

Worked answer

(x-2)^2(x+3)

Touch at 2 means (x-2)²

Cross at -3 means (x+3)

Problem 15

A graph falls on the left and falls on the right, and crosses the axis twice. What could its degree be?

Worked answer

Ends agree, so the degree is even

Falling both ways needs a negative lead

Degree 4 fits

Problem 16

For y=2(x-1)(x+2)², find the y-intercept.

Worked answer

-8

2(-1)(2)²

=2 × (-1) × 4 = -8

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