Teaching Algebra with Manipulatives

Chapter
7
Polynomial Functions
Teaching Notes and Overview
Mini-Project
Synthetic Substitution and Zeros
(p. 252 of this booklet)
Use With Lesson 7-4.
Objective Use synthetic substitution to find
zeros of a function.
This Mini-Project requires students to use
synthetic substitution to find the zeros of two
functions and to write a detailed explanation of
why synthetic substitution works. Ask students
to work in pairs.
Answers
1. 1, 0, 110;
1, 1, 2, 0;
1, 2, 3, 0;
1, 3, 2, 4;
1, 4, 1, 6;
1, 5, 6, 0;
No need to substitute 2; zeros are 3, 2,
and 1.
2. 1, 2, 1, 2, 0;
1, 1, 5, 9, 12;
1, 0, 7, 6, 0;
1, 1, 7, 1, 6;
1, 2, 5, 6, 0;
1, 3, 1, 3, 0;
No need to substitute 3; zeros are 3, 1, 1,
and 2.
3. See students’ explanations.
Algebra Activity
Exploring Iteration
(pp. 253–255 of this booklet)
Use With Lesson 7-7.
Objective Find and graph the first three
iterates of a function.
There are two parts to this activity:
• The first part deals with finding the first
three iterates of the function f(x) 1 2 x 5
for an initial value of x 0
2. Form groups of
two students. Have them read and write
down the work for each step. Discuss the
process and the first three iterates. Ask
students to complete Exercises 14.
• The second part of the activity focuses on
finding the first three iterates of the function
g(x) 4x for an initial value of x 0
0.25, and
then, drawing the graphical iteration. Before
you go over Activity 2, discuss the four basic
paths possible when a linear function is
iterated.
As a class, have students graph the function
g(x) 5x 7 and the function f(x) x on the
same set of axes. Then draw the graphical
iteration for x 0
1. State the slope of the linear
function and tell what type of path the
graphical iteration forms. The slope is 5
and the path staircases out.
Ask students to do the same for the function
g(x) 1 4 x 4. The slope is 1 4 and the path
staircases in.
After discussing these exercises, you may want
to talk about the process in general.
Step 1
Step 2
Step 3
Graph a function g(x) and the line
f(x) x on the coordinate plane.
Choose an initial value, x 0
, and locate
the point (x 0
,0).
Draw a vertical line from (x 0
,0) to the
graph of g(x). This will be the segment
from the point (x 0
,0) to (x 0
, g(x 0
)).
Step 4 Now draw a horizontal segment from
this point to the graph of the line
f(x) x. This will be the segment from
(x 0
, g(x 0
)) to (g(x 0
), g(x 0
)).
Repeat Steps 1–4 for many iterations.
Tell students that they may think of the line
f(x) x as a mirror that reflects each function
value to become the input for the iteration of
the function. The points at which the graph of
the function g(x) intersects the graph of the line
f(x) x are called fixed points. If you try to
iterate the initial value that corresponds to the
x-coordinate of a fixed point, the iterates will all
be the same.
Discuss the answers to Exercises 910. Ask a
student from each group to read what they
wrote for answers.
Algebra 2—Chapter x 7
© Glencoe/McGraw-Hill 249 Teaching Algebra with Manipulatives