MHF4U Polynomial Functions and Their Shapes: Notes and Practice

You already know two shapes very well: the straight line $y = m x + b$ and the parabola $y = a x^{2} + b x + c$ . They look completely different, but they belong to the same family. A line is a polynomial of degree 1, a parabola is a polynomial of degree 2, and if you keep going — degree 3, degree 4, degree 5 — you get curves with more wiggles but exactly the same personality: smooth, unbroken, defined for every real number, and ultimately controlled by just two pieces of information. This topic is about those two pieces of information: the degree and the leading coefficient.

What counts as a polynomial

A polynomial function is a sum of terms, each of which is a number multiplied by $x$ raised to a whole-number exponent. That single restriction — whole-number exponents only — is what rules out most of the other functions you know. There is no $x$ in a denominator (that would be a negative exponent), no $x$ under a root sign (that would be a fractional exponent), no $x$ in an exponent, and no trigonometric or logarithmic wrapper. Written with the exponents in decreasing order, a polynomial is in standard form. So $f (x) = x^{4} - 3 x^{2} + 7$ , $g (x) = 5 x^{3} - x$ and $h (x) = \frac{x ^{2} - 1}{4}$ all qualify (a constant denominator is just the coefficient $\frac{1}{4}$ ), while $y = \frac{3}{x}$ , $y = x$ , $y = 2^{x}$ , $y = ∣ x ∣$ and $y = sin x$ do not.

f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}, a_{n} \neq = 0, n \in W

The two numbers that decide everything

The degree

n

is the largest exponent on

x

. The leading coefficient

a_{n}

is the number attached to that highest-degree term — not the first number you happen to see written down. In

f (x) = 7 - 2 x^{3} + x^{2}

the degree is

3

and the leading coefficient is

- 2

. The constant term

a_{0}

f (0)

, which is the

y

-intercept.

AStandard form

f(x) = x³ − 2x² − 5x + 6
Degree and leading coefficient are visible immediately
y-intercept is the constant term: f(0) = 6
Zeros are hidden

BFactored form

f(x) = (x − 1)(x + 2)(x − 3)
Zeros are visible immediately: 1, −2, 3
Degree = sum of the exponents on the factors
Leading coefficient = product of the leading coefficients of the factors

Fig 1The characteristic shapes of polynomial graphs of degree 1, 2, 3 and 4.

End behaviour

End behaviour describes what the graph does far out on the left and far out on the right. Here is the reason it is so predictable: plug in a huge number like $x = 1000$ into $f (x) = 2 x^{3} - 50 x^{2} + 9$ . The cubic term gives $2000000000$ , while the $- 50 x^{2}$ term contributes about fifty million and the constant is negligible. The highest-degree term always wins eventually, so only the leading term matters at the ends. From there it is just two questions. Is the exponent even or odd? Is the coefficient positive or negative? An even power kills the sign of $x$ , so both ends go the same way; an odd power keeps the sign, so the ends go opposite ways.

Degree	Leading coefficient	As x → −∞	As x → +∞	Quadrants
even	$a > 0$	$y \to + \infty$	$y \to + \infty$	Q2 to Q1 (both ends up)
even	$a < 0$	$y \to - \infty$	$y \to - \infty$	Q3 to Q4 (both ends down)
odd	$a > 0$	$y \to - \infty$	$y \to + \infty$	Q3 to Q1 (down-left, up-right)
odd	$a < 0$	$y \to + \infty$	$y \to - \infty$	Q2 to Q4 (up-left, down-right)

The four end-behaviour cases. Even degree behaves like a parabola; odd degree behaves like a line.

Fig 2Even and odd degree with positive and negative leading coefficients — the four possible end behaviours.

VideoKhan Academy shows why the leading term takes over for very large and very negative x, which is the same degree and leading coefficient reasoning this section uses to predict where the ends of the graph point.Watch on YouTube

Domain, range and turning points

Every polynomial is defined for every real number — you can always multiply and add — so the domain is always ${x \in R}$ . The range depends on the parity of the degree. An odd-degree polynomial runs from $- \infty$ to $+ \infty$ , so its range is ${y \in R}$ . An even-degree polynomial has both ends pointing the same way, so it has an absolute minimum (if $a > 0$ ) or an absolute maximum (if $a < 0$ ), and its range is bounded on that side. A turning point is a place where the curve changes from rising to falling or vice versa. A polynomial of degree $n$ has at most $n - 1$ turning points and at most $n$ zeros. Note the words "at most": a cubic can have $2$ turning points or none at all, but never $3$ .

Finite differences

If you are handed a table of values with no equation, finite differences tell you the degree. Take the $y$ -values at equally spaced $x$ -values and subtract consecutive pairs to get the first differences. Subtract those to get the second differences, and keep going. For a polynomial of degree $n$ , the $n$ th differences are constant, and everything after that is zero. Better still, that constant is not random: it equals $a \cdot n!$ , where $a$ is the leading coefficient. So a constant third difference of $12$ means degree $3$ with $a = 12 \div 3! = 12 \div 6 = 2$ . (This works when the $x$ -values step by $1$ ; if they step by $h$ , the constant is $a \cdot n! \cdot h^{n}$ .)

Fig 3A difference table for the quadratic y = 2x² − 3x + 1: the second differences settle to the constant 4, which equals 2!·a, so a = 2.

Polynomial Functions and Their Shapes

Key things to remember

What counts as a polynomial

End behaviour

Domain, range and turning points

Finite differences

Worked examples

Where marks get lost

Flashcards 8

Practice