1. Graphing Calculator, Notebook
U7D1
Have Out:
Pencil, Highlighter,
Graphing Calculator, Notebook
Bellwork:
Answer the following questions in your packet.
CF – 1
Solve each of the following for x.
a) (x – 1) = 0
b) (x – 1)(x + 1) = 0
c) (x – 1)2 = 0
CF – 2
Sketch a graph of each of the following functions and indicate where each intersects the x–axis.
a) y = (x – 1)
b) y = (x – 1)(x + 1)
c) y = (x – 1)2

2. Solve each of the following for x.
CF – 1
Solve each of the following for x.
a) (x – 1) = 0
b) (x – 1)(x + 1) = 0
c) (x – 1)2 = 0
x = 1
x – 1 = 0
x + 1 = 0
x – 1 = 0
x = 1
x = –1
x = 1
CF – 2
Sketch a graph for each of the following functions and indicate where each intersects the x–axis.
a) y = (x – 1)
b) y = (x – 1)(x + 1)
c) y = (x – 1)2
Parabola
LINE
Parabola 5 –5 x y 5 –5 x y 5 –5 x y
(1, 0)
(–1, 0)
(1, 0)
(1, 0)

3. Definitions: Roots vs. Zeros
CF – 3
Definitions: Roots vs. Zeros
Roots:
solutions to an equation
For example, the roots of _________ are ______ and _____.
x2 – 9 = 0
x = –3
x = 3
Zeros:
the value of x when function f(x) = 0, that is, the locations where f(x) crosses the x–axis
For ex., the zeros of ______________ are ______ and _____.
f(x) = x2 – 3x – 4
x = –1
x = 4
CF – 4
Find the zeros for each of the following functions.
a) y = x2 – 6x + 8
b) y = x2 – 6x + 9
c) y = x3 – 4x
0 = x2 – 6x + 8
0 = x2 – 6x + 9
0 = x3 – 4x
0 = (x – 2)(x – 4)
0 = (x – 3)(x – 3)
0 = x(x2 – 4)
0 = x(x – 2)(x + 2)
x – 2 = 0 or x – 4 = 0
x – 3 = 0
x = 0 or x – 2 = 0 or x + 2 = 0
x = 3
x = 2, 4
x = 0, ±2

4. CF – 5
Using your graphing calculator, sketch the graph for the following functions.
a) y = (x – 1)2(x + 1)
b) y = (x – 1)2(x + 1)2
c) y = x3 – 4x
(–1, 0)
(1, 0)
(–1, 0)
(1, 0)
(–2, 0)
(2, 0)
(0, 0)
Parts (a) and (b) have examples of double zeros because the graphs “bounce” off the x–axis at these zeros.

5. Let’s review the basics:
CF – 6
Let’s review the basics:
3x2, y10, –2
Monomials: Expression that is a ___, ______, or any ___________. #
Example: _________
variable
combination
Example: _________
9x2 + 14
Binomial: _____ terms added or subtracted.
two
Trinomial: _____ terms added or subtracted.
three
Example: _________
x2 + 6x + 9
Polynomial: _____ terms added or subtracted.
many
Example: ____________
x3 – 5x2 + 3x – 1

6. Polynomials can be written in the form:
CF – 6
Polynomials can be written in the form:
There are several characteristics of polynomials:
coefficients
 The ____________ are _______ numbers.
real
 The _______ _______, a0, is not _______.
first
number
zero
 The _______ or ________ are __________ _________.
powers
exponents
positive
integers

7. The highest power of x in any of these terms is called the _________.
CF – 6
Essentially, a polynomial can be written as the sum of the terms in the form:
positive whole number
___________________
________________  x
any number
Ex: x3 – 5x2 + 3x – 1
The highest power of x in any of these terms is called the _________.
degree
Ex: x3 – 5x2 + 3x – 1
The degree is 3.
leading
coefficient
The _______ __________ is the coefficient of the term with the highest degree.
Ex: x3 – 5x2 + 3x – 1
The leading coefficient is 1.
descending
We always write polynomials in ____________ order.
Ex: –7x5 – 2x4 + 19x3 – x2 + 8x – 10
In the above example, the degree is ____ and the leading coefficient is ___. 5 –7

8. Examples: a) degree = 5 leading coefficient = 8 b) degree = 6
The following are examples of polynomial functions. Identify the degree of the polynomial and the leading coefficient.
Examples: a)
degree = 5
leading coefficient = 8 b)
degree = 6
leading coefficient = c)
degree = 3
leading coefficient = 7
Short cut: just count the number of factors with an “x”.

9. Examples: The following are NOT polynomial functions. Explain why not.
The exponent is a variable, but it’s supposed to be a whole number.
can be written as , which is a fractional exponent. b)
can be written as c)
Polynomials cannot have negative exponents.

10. Using a graphing calculator, work on the worksheet titled: “Graphs of Polynomial Functions”

11. Graphs of Polynomial Functions
Using your graphing calculator, sketch the graph of the following functions. a) b) c)
Degree:
Zeros:
Degree:
Zeros: 1
Degree:
Zeros: 2
none
x = ± 2

12. Graphs of Polynomial Functions
Using your graphing calculator, sketch the graph of the following functions. d) e) f)
Degree:
Zeros: 2
Degree:
Zeros: 3
Degree:
Zeros: 4
x = –3
x = –4,–1,0,3
x = ± 2, 3

13. Graphs of Polynomial Functions
Using your graphing calculator, sketch the graph of the following functions. g) h) i)
Degree:
Zeros: 3
Degree:
Zeros: 4
Degree:
Zeros: 3
x = –3,–1, 3
x = 0, 2
x = –4, –1, 2

14. a) How does the degree compare to the maximum number of real zeros?
The maximum number of zeros equals the degree.
b) What do you notice about the shapes of the graphs for even–degree polynomial functions and odd–degree polynomial functions?
Anything even degree is similar to a parabola’s ends.
Anything odd degree is similar to cubic’s ends.
c) The _______ and ___________________ determine the graph’s ______________.
degree
leading coefficients
end behavior

15. Even degree
+ leading coefficient
Even degree
– leading coefficient
Odd degree
+ leading coefficient
Odd degree
– leading coefficient

16. Do not move until everyone’s name is called.
It’s time to
Change Seats!
Do not move until everyone’s name is called.