Lesson 2.2
Read: Pages 116-126
Page 112: #1-9 (EOO), 15-43 (EOO),
59-75 (EOO)
Polynomial Functions of Higher
Degree
f (x)  a(x  h)  k
2

Objective
Students will use transformations to sketch and analyze graphs of polynomial
functions, use the Leading Coefficient Test to determine end behavior, and
find and use zeros of polynomial functions as sketching aids.
Characteristics and Graphs of Polynomial
Functions
Graph the Following
• 𝑓 𝑥 = −𝑥 5
• 𝑔 𝑥 = 𝑥 4 +1
• ℎ 𝑥 = (𝑥 + 1)4
Use the Leading Coefficient Test to describe the right-hand
and left-hand behavior of each polynomial function.
𝑓 𝑥 = −𝑥 3 + 4𝑥
𝑔 𝑥 = 𝑥 4 − 5𝑥 2 + 4
ℎ 𝑥 = 𝑥5 − 𝑥
Zeros of Polynomial Functions
A Polynomial f of degree n has at most:
1. n real zeros.
2. n-1 relative minima and maxima.
If f is a polynomial function and a is a real number,
the following statements are equivalent.
1)
x = a is a zero of the function f
2) x = a is a solution of the polynomial equation f(x) = 0
3) (x − a) is a factor of the polynomial f(x)
4) (a, 0) is an x-intercept of the graph of f
Find all zeros of: 𝑓 𝑥 = 𝑥 3 − 𝑥 2 − 2𝑥.
𝑓 𝑥 = 𝑥 3 − 𝑥 2 − 2𝑥
0 = x(𝑥 2 − 𝑥 − 2)
0 = x(𝑥 − 2)(𝑥 + 1)
(factors)
So, the real zeros x = 0, x = 2, and x = -1.
Corresponding x-intercepts are (0,0), (2,0),
(-1,0).
Repeated Zeros
For a polynomial function, a factor of
(𝑥 − 𝑎)𝑘 , 𝑘 > 1, yields a repeated zero 𝑥 = 𝑎 of
multiplicity k.
1. If k is odd, the graph crosses the x-axis at x
= a.
2. If k is even, the graph touches the x-axis (but
does not cross the x-axis at x = a.
Find polynomial functions with the following zeros.
(There are many correct solutions.)
a. -
1
, 3, 3
2
𝑓 𝑥 = (2𝑥 + 1)(𝑥 − 3)2
𝑓 𝑥 = 2𝑥 + 1 𝑥 2 − 6𝑥 + 9
𝑓 𝑥 = 2𝑥 3 − 11𝑥 2 + 12𝑥 + 9
Sketch the graph of 𝑓 𝑥 = 3𝑥 3 − 4𝑥 2 by hand.
1. Apply the Leading Coefficient Test. Because the leading
coefficient is positive and the degree is odd, you know that
the graph eventually drop to the left and rise to the right
2. Find the Real Zeros of the Polynomial. By factoring,
𝑓 𝑥 = 3𝑥 3 − 4𝑥 2 = 𝑥 2 (3𝑥 − 4)
3. Plot a few additional points.
4. Draw the Graph.