Precalculus (A)
Name: _________________________
2.2: Polynomial Graphs – Self-Guided Notes
n–1
2
Recall:
A polynomial
function
is aisfunction
of the
form
f(x)f(x)
= a=nxann+
Recall:
A polynomial
function
a function
of the
form
xn a+n –a1nx– 1xn –+1 …
++
… a+2xa2+
x2 a+1xa+
1x a0,
where
is a nonnegative
integer and
an, aand
real numbers
and an ≠and
0. an ≠ 0.
n – 1,aa
+ a0n, where
n is a nonnegative
integer
, –a2n,–etc…
are real numbers
nn
1, an –are
2, etc…
Fill in the blanks below:
A polynomial of degree n has at most _______ distinct real zeros and at most _________
turning points.
Example: f(x) = 4x6 – 3x3 + 8 is of degree _____. It has at most ______ distinct real zeros
and at most _______ turning points.
Exploration:
The purpose of this exploration is to examine the role of the exponent on each factor and its effect on the
graph of the polynomial.
Directions
Using your calculator, make a quick sketch of the graph of each of the following functions.
Use the following viewing window: Xmin:-3 Xmax: 3.8 Ymin:-100 Ymax: 100
Mark the x- and y-intercepts with their values. Do not worry about scaling your drawing.
Y2 has been done for you as an example.
y1  (x  2)(x  1)(x  3)
y2  (x  2)2 (x  1)(x  3)
Degree y1 =
(-2, 0)
y3  (x  2)2 (x  1)4 ( x  3)
Degree y3 =
Degree y2 = 4
(0, 12)
(1, 0) (3, 0)
y 4  (x  2)3 (x  1)5 (x  3)2
Degree y4 =
Under what circumstances did you find a “pass through” point? “Pass through would look like either of
these:
or
Under what circumstances did you find a “bounce” point? By “bounce” we mean it looks like this:
The degree of a factor of a polynomial is called the multiplicity of the factor. For each function y1
through y6, state each zero along with its multiplicity.
*Example: y1: -2 has multiplicity 1, 1 has multiplicity 1, 3 has multiplicity 1.
Y2:__________________________________________________
Y3:__________________________________________________
Y4:__________________________________________________
Examples: For each polynomial that follows: a) determine the end behavior by using limits
b) determine the zeros and their multiplicities (factor if you need to)
c) find the y-intercept and sketch a possible graph
1) f ( x )  x  x  1
2
 x  4
2) g(x )   x  1
2
 x  3 2  x 
3
Example 3: For each polynomial that follows: a) determine the end behavior by using limits
b) determine the zeros and their multiplicities (factor if you need to)
c) find the y-intercept and sketch a possible graph
3) h(x)  3x 3  6x 2  3x
Questions 4 & 5: Find a polynomial function of degree n with only the following real zeros. (More than one answer is
possible)
4) real zeros: -5, 4; n = 4
5) real zeros: 2, 1, 4; n = 5
Questions 6 – 8: Define a function of least degree that satisfies the conditions. (it may help to sketch the graph first)
6) zeros: -3, -1, 3
2 turning points
y-intercept is (0, -45)
7) zeros: -3, 1, 5
3 turning points
concave up on (-∞, 1) (4, ∞)
concave down on (1, 4)
lim f (x)   , lim f (x)  
x 
8)
(0, 6)
x 