Polynomial Functions
Equations
and Graphs
The Factor
Theorem
Characteristics
The
Remainder
Theorem
http://www.purplemath.com/modules/polyends5.htm
Math 30-1
1
A cross-section of a honeycomb
has a pattern with one hexagon
surrounded by six more
hexagons. Surrounding these is
a third ring of 12 hexagons, and
so on.
The quadratic function h (r) that models the total number of
hexagons in a honeycomb, where r is the number of rings is
h(r )  3r  3r  1
2
This could also be
referred to as a
Polynomial Function.
Math 30-1
2
Identifying Polynomial Functions
Any function in the form:
f(x) = anxn + an-1xn - 1 + an - 2xn -2 + … + a1x + a0
where
• x is a variable
• the coefficients of the variable, an , an - 1 …. a0 , are real numbers
• n is a whole number
Circle only the polynomial functions.
f ( x)  x  3x  2
2
y=
4x-3
+
8x2
f ( x)  x  3x  6
+ 3x - 2
5
y= 2
2
f ( x) 
3 3
x  9 x 2  10 x
4
3x 2  6x  10
y
2x
3
Explain why the other functions are not polynomials.
f(x) = anxn + an-1xn - 1 + an - 2xn -2 + … + a1x + a0
f ( x)  x  5 x  5 x  5 x  6
4
3
2
• The degree of the polynomial function is n,
degree of 4
the exponent of the greatest power of x.
relates to the number of changes of direction on the graph of the function
relates to the direction of opening or end behaviour of graph
• The leading coefficient is an,
+1
the coefficient of the greatest power of x.
relates to the direction of opening or end behaviour of graph
• The constant term is a0, since x0.
relates to the y-intercept of the graph of the function
Math 30-1
-6
4
Characteristics of Polynomial Functions
Odd Degree Polynomial Functions
Degree 1
Linear
y = ax + c
One direction
a>0
a<0
Leading coefficient positive
Leading coefficient negative
End behaviour - +
End behaviour + -
Constant term c = y-intercept
Constant term c = y-intercept
x-intercepts:
x-intercepts one
Max or Min?
one
Max or Min ?
Math 30-1
5
Even Degree Polynomial Function
Degree 2
Quadratic
y = ax2 + bx + c
Two directions
a>0
a<0
Leading coefficient positive
Leading coefficient negative
End behaviour + +
End behaviour -
Constant term c = y-intercept
Constant term c = y-intercept
x-intercepts:
x-intercepts:
Max or Min ?
none, one, two
Max or Min ?
Math 30-1
-
none, one, two
6
Your Turn: Graph the given polynomial function and identify the characteristics
y = ax3 + bx2 + cx + d
Cubic
Degree ____ Odd or Even?
_________ possible changes in direction
a>0
yx
3
a<0
y  x  2x  x  2
3
2
y  x
3
Leading coefficient
Leading coefficient
End behaviour
Resembles
End behaviour
Resembles
Constant term
Domain
Constant term
Domain
Range
x-intercepts:
x-intercepts:
Max or Min?
Max or Min?
Math 30-1
y   x3  2 x 2  x  2
Range
7
Your Turn: Graph the given polynomial function and identify the characteristics
Quartic y = ax4 + bx3 + cx2 + dx + e Degree ____ Odd or Even?
_________ possible directions
a>0
y  x4
a<0
y  x 4  5 x3  5 x 2  5 x  6
y   x4
y   x 4  5 x3  5 x 2  5 x  6
Leading coefficient
Leading coefficient
End behaviour
Resembles
End behaviour
Resembles
Constant term
Domain
Constant term
Domain
Range
x-intercepts:
x-intercepts:
Max or Min?
Max or Min?
Math 30-1
Range
8
Your Turn: Graph the given polynomial function and identify the characteristics
Quintic y = ax5 + bx4 + cx3 + dx2 + ex + f
Degree __ Odd or Even
_________ possible directions
a>0
a<0
y  x5 y  x 5  3x 4  5 x 3  15 x 2  4 x  12 y   x5 y  x 5  3x 4  5 x 3  15 x 2  4 x  12
Leading coefficient
Leading coefficient
End behaviour
Resembles
End behaviour
Resembles
Constant term
Domain
Constant term
Domain
Range
x-intercepts:
x-intercepts:
Max or Min?
Max or Min?
Math 30-1
Range
9
Use the following information to answer the next four questions
Which graph could represent a polynomial whose leading term
is –3x4 +… ?
B
Which graph could represent a polynomial whose leading term
is 3x4 +… ?
D
Which graph could represent a polynomial whose leading term
is 4x3 +… ?
A
Which graph could represent a polynomial whose leading term
is -4x3 +… ?
C
Math 30-1
10
Identify the following characteristics for each polynomial
function:
• the type of function and whether it is of even or odd degree
• the end behaviour of the graph of the function
• the number of possible x-intercepts
• whether the function will have a maximum or minimum value
• the y-intercept
g (x) = -x3 + 8x2 + 7x - 1
f (x) = x4 + x2 - x + 10
Cubic function, degree 3, odd
Quartic function, degree 4, even
End behaviour +
End behaviour +
-
+
At most 3 x-intercepts
At least one x-intercept
At most 4 x-intercepts
At least no x-intercepts
Neither max nor min
Absolute minimum value
y-intercept at -1
Math 30-1
y-intercept at 10
11
The height, h, in metres, above the ground of an object dropped
from a height of 60 m is related to the length of time, t, in seconds,
that the object has been falling. The formula is h = -4.9t2 + 60.
a) What is the degree of this function?
b) What are the leading coefficient and constant of this function?
c) What does the constant represent?
d) What are the restrictions on the domain of the function?
e) Describe the end behaviour of the graph of this function.
f) Use the formula to determine how long an object will take to hit
the ground if it is dropped from a height of 60 m (nearest tenth of
a second).
Page 114
1, 2, 3, 5, 6, 9, C4
Math 30-1
12