MHF 4UI-Polynomial Functions
A function is a relationship between two sets of numbers in which each value of x, in the
________________, corresponds to exactly one value of y, in the __________________.
f(x) is called function notation and represents the y value of the function for some given value of
x.
Polynomial Functions:
A polynomial function is a function whose equation is defined by a polynomial in one variable.
f ( x)  a n x n  a n 1 x n 1  a n 2 x n  2  ...  a 2 x 2  a1 x  a0
where a0 , a1 , a 2 ,...a n are real
numbers and n is a natural number.
The coefficient of the highest term is called the __________________________.
The ___________ of the polynomial function is the value of the highest exponent of the variable.
A ____________ function, f ( x)  ax  b , is a polynomial function of degree____________.
A ____________ function, f ( x)  ax 2  bx  c , is a polynomial function of degree________.
A __________ three polynomial function , f ( x)  ax 3  bx 2  cx  d , is called_____________.
Increasing and Decreasing Functions
A function f, is increasing on an interval I if the value of f(x) increases as the value of x
increases. f ( x1 )  f ( x2 ) when x1  x2 and x1 and x2 are in I.
A function f, is decreasing on an interval I if the value of f(x) decreases as the value of x
increases. f ( x1 )  f ( x2 ) when x1  x2 and x1 and x2 are in I.
Local Minimum and Local Maximum values
A turning point is a point on a curve that is
higher or lower than all nearby points.
A turning point occurs wher a function
Changes from increasing to decreasing or
vica versa. A local ____________ occurs
when the function changes from increasing
to decreasing. A local _____________
occurs when the function changes from
decreasing to increasing.
Zeros of a Polynomial Function
The zeros of a function are the ___________________, and can easily be found when the
polynomial function is in __________ form. If k represent the leading coefficient; the linear
function f ( x)  k ( x  s ) has a zero ___; the quadratic function f ( x)  k ( x  s)( x  t ) has
zeros ________; the cubic function f ( x)  k ( x  s)( x  t )( x  u ) has zeros _____________.
[s,t and u can be any real value]
End Behaviour of a Polynomial Function
An important property of a function is its end behaviour.
As x takes on a very large positive value “ x   ” and as x takes on a very large negative value
“ x   ” we observe the values of the function f(x) get larger in a positive or negative
direction.
6
As x   f(x) gets larger in a negative
direction _____________.
4
2
-10
-5
5
10
As x   f(x) gets larger in a positive
direction _____________.
-2
-4
-6
Even and odd functions
A function is an even function if f(-x) = f-x) for all values of x. An even function is symmetrical
about the ___________.
A function is odd if f(-x) = -f(x) for all value of x. An odd function is symmetrical about the
_________.
Note: There are functions which are neither even nor odd.
Finite Differences in Polynomial Functions
The nth finite differences of a polynomial function of degree ___ are ________.
The third finite difference of a degree - _______ polynomial is related to the leading coefficient
by a factor of __________________.
The fourth finte difference of a degree-_________ polynomial is related to the leading
coefficient by a factor of ____________________.