Polynomial Functions - Introduction Worksheet
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1
2
3Overall degree of n
Linear term
Constant term
1.5A – Polynomial Functions Introduction
n
n-1
n-2
1
Standard form:
f(x) = a
x
+a
x
+ a
x
+ … + a
x
+ a
n
n-1
n-2
1
0
k is a constant
Factored form:
f(x) = k(x - a)(x - b)(x - c)(x + d)
that is used to
represent a
To sketch the graph of a polynomial function one will need to identify its’ key features;
family of
functions that all
have the same
a) general shape - from overall degree of function
zeros
b) direction of opening - sign in front of highest degreed term
c) the zeros (where curve crosses x-axis) - determined form factored form
Make notes on the following chart to outline some general observations when graphing the
following polynomial functions on a graphing calculator
Type of
Example
Factored
# of Zeros
Other notes
Degree
function
linear
1
f(x) = 2x + 4
f(x) = 2(x + 2)
0 or 1
1 positive section
2
quadratic
2
g(x) = -x
- 5x - 6
g(x) = -(x + 2)(x + 3)
0, 1 or 2
Opens down
3
2
cubic
3
h(x) = x
– 2x
h(x) = xx(x – 2)
1, 2, or 3
Touch point at x = 0
4
3
2
2
quartic
4
m(x) = x
– 4x
– 12x
m(x) = x
(x - 6)(x + 2)
0 to 4
4 sections
2
fifth
5
f(x) = ?
f(x) = (x-2)
(x+3)(x-1)(x+2)
1 to 5
Touch point at x = 2
2
2
sixth
6
g(x)
g(x)= x
(x-3)
(x-1)(x+2)
0 to 6
6 possible sections
Example 1:
Sketch the following polynomial functions;
2
a) g(x) = (x - 3)
(x + 1)
opens: does not apply to odd degreed functions
rd
degree: 3
(so 3 possible sections starting with positive)
x = -1 (crosses x-axis)
x = 3 (touch point on x-axis)
2
b) f(x) = -2x(3x+1)(x-3)
open: down
th
degree: 4
(so 4 possible sections)
zeros: crosses at x = -1/3, x = 0
touches at x = +3
To find zeros set each
factored bracket to zero.
So
3x + 1 = 0
3x = -1
x = -1/3
Example 2: Determine a family of functions given the following graph
a)
Zeros are at x = -5, +1, + 5
g(x) = k (x + 5)(x – 1)(x – 5)
Need to use opposite sign
when write in factored
form. Why?
How could you use
max at (-2, 6) to find
value of k?
1.5A – polynomial functions Introduction
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