A Polynomial function is a function in which the independent variable is a non-negative integer exponent.

An algebraic term with the highest exponent is the leading term that determines the degree of the function; its coefficient is the leading coefficient.

## Polynomial Functions: Characteristics and Graphs

A degree of a polynomial function = maximum number of x-intercepts.

A turning point is where a function changes from increasing to decreasing or vice versa.

The maximum number of turning points (local maximums and local minimums) = one less than the degree of a function.

The sign of the leading coefficient determines the end behaviour of a polynomial function:

• Odd number degree + positive leading coefficient: Q3 – Q1
• Odd number degree + negative leading coefficient: Q2 – Q4
• Even number degree + positive leading coefficient: Q2 – Q1
• Even number degree + negative leading coefficient: Q3 – Q4

In the expanded form, the constant value = y-intercept of the function.

In the factored form, the factors help determine the x-intercepts (roots).

The order of x-intercepts (roots) is how many times that particular x-intercept is repeated.

If the x-intercept is of an even number order (2,4,6,…) – there is a bounce and a turning point at that x-intercept.

If the x-intercept is of an odd number order (1,3,5,…) – the graph curves through the x-intercept and there is no turning point at that x-intercept.

## Polynomial Functions: Symmetry

Odd Function – odd symmetry – when a function is symmetrical about the origin (rotational symmetry).

Even Function – even symmetry – when a function is symmetrical about the y-axis (mirror symmetry).

Neither Even nor Odd Function – neither symmetrical about the origin nor about the y-axis.

It is also possible to determine the symmetry algebraically, by testing the equation of the function.

Given the equation of the function $f(x)$, we can test $f(-x)$ and see if every term changes the sign to the opposite or not.

For example, the function  $f(x)=3x^3-x$  is ODD, since

$f(-x)=3(-x)^3-(-x)$

$f(-x)=-3x^3+x$

Every sign within the function equation has changed to the opposite.

The function  $f(x)=2x^4-3x^2+1$  is EVEN, since

$f(-x)=2(-x)^4-3(-x)^2+1$

$f(-x)=2x^4-3x^2+1$

Every sign within the function equation has remained unchanged.

The function  $f(x)=3x^3-2x^2-15$  is NEITHER, since

$f(-x)=3(-x)^3-2(-x)^2-15$

$f(-x)=-3x^3-2x^2-15$

The signs of some terms have changed and others – remained the same.

NOTE: It is important to differentiate between odd/even degree and odd/even symmetry – those are different concepts, even though similar terminology is used to describe them.

Check your understanding with Polynomial Functions Quiz and Even and Odd Symmetry Quiz