Functions
Name____________________
Lesson 4.5
Topic _________________________________________________
Activity 1. Read p. 236 and answer the following questions.
The
Location
Principle
Activity 2. Study the following example and do 1, 2, 3, 4, 5 6 (worksheet)
and 13, 15, 17 on p. 240
Example 1
Determine between which consecutive integers the real zeros of f(x) = x3
+ 3x2 - 4x + 6 are located.
There are three complex zeros for this function. According to Descartes’ Rule of Signs,
there are two or zero positive real roots and one negative real root. You can use
substitution, synthetic division, or the TABLE feature on a graphing calculator to evaluate
the function for consecutive integral values of x.
Use the TABLE feature.
The change in sign between -5 and -4 indicates that a zero exists between -5 and -4. This result is
consistent with Descartes’ Rule of Signs.
Determine between which consecutive integers the real zeros of each
function are located.
1.
The zero(s) exists between ___________________________________________.
2.
The zero(s) exists between ___________________________________________.
3.
The zero(s) exists between ___________________________________________.
4.
The zero(s) exists between ___________________________________________.
5.
The zero(s) exists between ___________________________________________.
6.
The zero(s) exists between ___________________________________________.
13.
The zero(s) exists between ___________________________________________.
15.
The zero(s) exists between ___________________________________________.
17.
The zero(s) exists between ___________________________________________.
Activity 3. Study the following example and do 7, 8, 9, 10, 11, 12
(worksheet) and 19, 21, 23 on p. 240
Example 2
Approximate the real zeros of f(x) = 5x3 - 2x2 - 4x + 1 to the nearest tenth.
There are three complex zeros for this function. According to Descartes’ Rule of Signs,
there are two or zero positive real roots and one negative real root.
Use the TABLE feature of a graphing calculator. To find the zeros to the nearest tenth, use
the TBLSET feature changing Tbl to 0.1. There are zeros between -0.9 and -0.8, between
0.2 and 0.3, and at 1.
Since 0.36 is closer to zero than -0.665, the zero
is about -0.8.
Since 0.16 is closer to zero than -0.245, the zero
is about 0.2.
The third zero occurs at 1.
Approximate the real zeros of each function to the nearest tenth.
7.
Real Zeros ___________________________________
8.
Real Zeros ___________________________________
9.
Real Zeros ___________________________________
10.
Real Zeros ___________________________________
11.
Real Zeros ___________________________________
12.
Real Zeros ___________________________________
19.
Real Zeros ___________________________________
21.
Real Zeros ___________________________________
23.
Real Zeros ___________________________________
Activity 4. Read p. 238 and answer the following questions.
1. What is the upper bound? ______________________________
Upper
Bound
Theorem
2. 1. What is the lower bound? ______________________________
Lower
Bound
Theorem
Activity 5. Study the following example and do 13, 14, (worksheet) and 27,
29, 31 on p. 241.
Example 3
Use the Upper Bound Theorem to find an integral upper bound and the Lower
Bound Theorem to find an integral lower bound of the zeros of
f(x) = x3 + 5x2 - 2x - 8.
The Rational Root Theorem tells us that 1, 2, 4, and 8 might be roots of the
polynomial equation x3 + 5x2 - 2x - 8 = 0. These possible zeros of the function are good
starting places for finding an upper bound.
f(x) = x3 + 5x2 - 2x - 8
r
1
5
-2
-8
1
1
6
4
-4
2
1
7 12 16
f(-x) = -x3 + 5x2 + 2x - 8
r
-1
5
2
-8
1
-1
4
6
-2
2
-1
3
8
8
3
-1
2
8 16
4
-1
1
6 16
5
-1
0
2
2
6
-1
-1
-4 -32
An upper bound is 2. Since 6 is an upper bound of f(-x), -6 is a lower bound of f(x). This
means that all real zeros of f(x) can be found in the interval -6  x  2.
13.
Interval ______________________
14.
Interval ______________________
27.
Interval ______________________
29.
Interval ______________________
31.
Interval ______________________