Unit 6 A: Polynomial Functions
Name: _______________________
Period: ______________________
Section 6.1 Analyzing Graphs of Polynomial Functions
Objective(s): Analyze graphs of polynomial functions to determine its characteristics.
Essential Question: Explain which term of the polynomial function is most important when determining
the end behavior of the function.
Homework: Assignment 6.1. #1 – 15 in the homework packet.
Notes:
The graph of y = x² is a _________________.
It’s graph is
The graph of y = - x² is
Higher Degree Polynomials
y = x³ + 4x² - x – 2 or y = 2x4 + 3x³ + 2x² + 3x – 1 are called higher degree polynomials. Notice that
their degree (highest power) is more than 2.
All of these graphs will be curves with high and/or low points called turning points (also called local
maximum or local minimum. The graphs are “snake-like”. Examples are shown below.
A
B
How many turning points are there in
C
D
Graph A?_________Graph B? __________
Graph C? ______________ Graph D?________________
Reflection:
1
Turning Points
The graph of a polynomial function has AT MOST n – 1 turning points, where n = the degree (highest
power of x). It may have fewer turning points.
For the examples, find the maximum number of turning points for each:
Example 1:
f(x) = x3 + 6x2 + x – 1
_________
Example 2:
f(x) = 2x10 – 8x4 + 3x
_________
End Behavior for Even Degree Polynomial Functions
The graph of y = x²
shows end behavior of rising (going up) on both ends.
The graph of any parabola y = ax² + bx + c will have the same shape, as long as a is positive.
If a is NEGATIVE, the graph of y = ax² + bx + c will have
shape.
Both ends of the graph are going down (falling).
If the degree of the polynomial function is EVEN, it will have the same end behavior as y = ax². Going
up (rising) on both ends, or going down (falling) on both ends. There may be turning points in
between. They will basically be a “ U ” or “” shape with wiggles (turning points) possibly in the
middle. Look at these examples
To Review: Degree – even
a) Leading coefficient positive
Reflection:
b) Leading coefficient negative
2
End Behavior for Odd Degree Polynomial Functions
If the degree of the polynomial function is ODD and LEADING COEFFICIENT IS POSITIVE it will have the
shape shown.
The left side is falling and the right side is rising. The sides are going in opposite direction. Remember
there may be turning points in between.
If the degree of the polynomial function is ODD and LEADING COEFFICIENT IS NEGATIVE (-) it will have
the shape shown. The sides are going in opposite direction. Remember there may be turning points in
between.
To Review: Degree – odd
a) Leading coefficient positive
b) Leading coefficient negative
How many turning points does Graph A have?__________Graph B __________
End Behavior for Polynomial Functions
The end behavior is described in mathematical language with symbols like the → arrow, which means
“approaches”, and the symbol for infinity ∞.
When we write x → + ∞ it means the right side of the graph.
When we write x → - ∞ it means the left side of the graph.
If a graph is rising (going up), we say
f(x) → + ∞
If a graph is falling (going down), we say
f(x) → - ∞
The answer to the end behavior of any polynomial function is
either + ∞ OR - ∞
Reflection:
3
Example 3: What is the end behavior of f(x) as x → + ∞ (on the right) for the graph shown below?
Remember, the answer to end behavior has to be + ∞ or - ∞.
f(x) → ______
f(x) → ______
f(x) → ______
Example 4: What is the end behavior of f(x) as x → - ∞ (on the left) for the graphs shown above?
(Looking at the left side)
f(x) → ______
f(x) → ________
f(x) → _________
Finding the end behavior of f(x) given the polynomial (but no graph).
If you are given the polynomial function, you will need to think what the sketch of the graph would look
like. Remember that this depends on the degree and if the leading coefficient is positive or negative.
(see pages 2 – 3)
SKETCH and describe the end behavior of f(x) as
x→+ ∞
(i.e. Is the graph going up or down on
the right side?)
Sketch here
Example 5:
f(x) = 3x4 – 4x2 + 1
f(x) → _________
Example 6:
f(x) = 2x5 + x2
f(x) → _________
Example 7:
f(x) = -x7 + 5x4 – x + 2
f(x) → _________
Example 8:
f(x) = -x6 + 9x4 + x3
f(x) → _________
Reflection:
4
Describe the end behavior of f(x) as
x→- ∞
(i.e. Is the graph going up or down on the left
side?)
Example 9:
f(x) = -2x2 – 4x + 7
f(x) → _________
Example 10:
f(x) = -8x3 + 6x2 – 12x – 9
f(x) → _________
Example 11:
f(x) = -x5 + 9x4 – 3x2 + 22
f(x) → _________
Example 12:
f(x) = 8x4 + x2 + 9
f(x) → _________
Vocabulary Connection: The following statements are equivalent.
- Zero: k is a zero of the function f(x).
- Factor: (x – k) is a factor of the function f(x).
- Solution: k is a solution of the equation f(x) = 0
- x-intercept: if k is a real number, then k is an x-intercept of the graph of the function f(x).
Example 13:
If 7 is a zero of f(x) = x3 – 7x2 + 2x – 14, then
Factor =
Solution =
x-intercept = (
, 0)
Example 14:
If -2 is a zero of f(x) = x3 + 3x2 + x – 2, then
Factor =
Solution =
x-intercept = (
, 0)
Reflection:
5
Section 6.2 Graphing a Polynomial in Factored Form
Objective(s): Graph polynomial functions.
Essential Question: What is the greatest number of local maximums and minimums that a cubic
function can have?
Homework: Assignment 6.2. #16 – 21 in the homework packet.
Steps to Graph
1. If the polynomial function is in factored form, place points on the x-axis where the x-intercepts
occur.
2. Determine the shape by deciding on the degree.
3. If a factor is raised to an EVEN power, example (x – 3)² the graph will not cross the x-axis at 3. It
will TOUCH at 3 on the x-axis and then turn and go the other way.
4. Sketch the graph.
Graph the function. Describe the end behavior of f(x).
Example 1:
f(x) = (x – 2)(x + 3)(x – 6)
Describe the end behavior of f(x) as
x→+ ∞
f(x) → _________
Describe the end behavior of f(x) as
f(x) → _________
Reflection:
Example 2:
f(x) = x(x + 2)(x + 4)(x + 6)
Describe the end behavior of f(x) as
x→+ ∞
f(x) → _________
x→- ∞
Describe the end behavior of f(x) as
x→- ∞
f(x) → _________
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Example 3:
f(x) = -(2x + 1)(x – 4)(x – 5)(x + 3)
Describe the end behavior of f(x) as
x→+ ∞
f(x) → _________
Describe the end behavior of f(x) as
Example 4:
f(x) = -(x + 4)(x – 2)(2x + 3)
Describe the end behavior of f(x) as
x→+ ∞
f(x) → _________
x→- ∞
Describe the end behavior of f(x) as
x→- ∞
f(x) → _________
f(x) → _________
Example 5: f(x) = (x – 1)² (x + 3) (x+4)
Example 6: f(x) = x(x – 1)² (x – 4)(x +2)
Describe the end behavior of f(x) as
Describe the end behavior of f(x) as
x→+ ∞
f(x) → _________
Describe the end behavior of f(x) as
f(x) → _________
Reflection:
x→+ ∞
f(x) → _________
x→- ∞
Describe the end behavior of f(x) as
x→- ∞
f(x) → _________
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Sample CCSD Common Exam Practice Question(s):
1. Which best represents the graph of the polynomial function y   x 4  2 x 2  2 x  3 ?
2. Which describes the end behavior of the graph of f  x   x 4  5 x  2 as x   ?
A.
f  x   
B.
f  x   
C.
f  x  0
D.
f  x  2
3. Use the graph of a polynomial function below.
What are the zeros of the polynomial?
Reflection:
A.
{2}
B.
{2}
C.
{3, 1, 4}
D.
{3,  1,  4}
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Section 6.3 Using Properties of Exponents
Objective(s): Use properties of exponents to simplify and evaluate expressions.
Essential Question: Which properties of exponents require you to check that two or more bases are the
same before applying the property?
Homework: Assignment 6.3. #22 – 37 in the homework packet.
Notes:
How many factors of 2 are there in the product 23  25 ? Use your answer to write the product as a single
power of 2.
Write each product as a single power.
22  25 =
33  36 =
71  7 6 =
x4  x4 =
Write each quotient as a single power of 2 by first writing the numerator and denominator in “expanded
form” (for example, 23  2  2  2 ) and then canceling common factors.
23
=
21
26
=
22
23
=
27
Properties of Exponents
a m  a n  a mn
(ab)m  a mbm
am
 a mn
n
a
am
a

 
bm
b
(a m )n  a mn
a0  1
am 
m
1
am
Reflection:
9
Simplify. Write the answer with positive exponents.
Example 1:
7 
3 4
Example 2:
 3x y 
2
4 3
Example 3:
y6
y8
Example 4:
4x 0
Example 5:
43
Example 6:
6x 4
Example 7:
 r 
 5
s 
2
Example 8:
3x 2 y 4 z 8
6 xy 6 z 2
Example 9:
6 x15 y12 z 4
9 x 7 y 2 z 4
Reflection:
10
Section 6.4 Adding, Subtracting, and Multiplying Polynomials
Objective(s): Simplify polynomial expressions.
Essential Question: What is the advantage of the box method when multiplying polynomials?
Homework: Assignment 6.4. #38 – 52 in the homework packet.
Notes:
Perform the indicated operation.
Example 1:
(10x3 – 5x + 8) + (-6x2 + 12x + 3)
Example 2:
6x – (14 – 4x)
Example 3:
(-4x2 + 5) – (-x3 – 10x2 – 4)
Multiply
Example 4:
-10x(6x – 8)
Example 5:
(3x – 10)(x – 6)
Example 6:
(2x – 3y)2
Example 7:
(x + 8y)2
Example 8:
(6x – 4)(2x2 + 7x – 9)
Reflection:
11
Example 9:
(2x + 4)(x2 – 3x – 1)
Example 10:
(2x – 3)(x – 7)(x + 5)
Sample CCSD Common Exam Practice Question(s):
1. Which polynomial represents the product of  x  2   x 3  8  ?
A. x 4  2 x3  8 x  16
B. x 4  2 x3  8 x  16
C. x 4  16
D. x 4  16
Reflection:
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Section 6.5 Factoring and Solving Polynomial Equations
Objective(s): Solve polynomial equations by factoring.
Essential Question: Give an example of a binomial that can be factored either as the difference of two
squares or as the difference of two cubes. Show the complete factorization of your binomial.
Homework: Assignment 6.5. #53 – 70 in the homework packet.
Notes:
General Steps to Factor
1. Factor out the GCF first (if possible)
2. Binomial: Difference of squares
a2 – b2 = (a + b)(a – b)
3. Binomial: Sum/Difference of cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
4. Trinomial: Use ac method, box method, etc.
5. Four terms: Factor by grouping
Factor the polynomial completely.
Example 1:
18x3 – 6x2y + 10xz3
Example 2:
3x + 21 + xy + 7y
Example 3:
x2 – 9x + 14
Example 4:
x2 – 3x – 88
Example 5:
x2 + x – 20
Example 6:
x2 – 49
Reflection:
13
Example 7:
9x2 – 25y2
Example 8:
3x2 – 75
Example 9:
7x2 – 5x – 2
Example 10:
6x2 – 26x – 20
Example 11:
x4 – 7x2 + 6
Example 12:
x3 + 27
Example 13:
8x3 – 125
Example 14:
27x3 + 1
Example 15:
2y3 + 54z3
Reflection:
14
Solve the equation.
Example 16:
2(x – 4)(x + 5) = 0
Example 17:
x2 + 3x - 40 = 0
Example 18:
x2 + 7x = 60
Example 19:
2x3 + 18x2 + 28x = 0
Example 20:
x3 – x = -8x2 + 8
Example 21:
x3 + 5x2 = 50x
Example 22:
x4 – 18x2 + 32 = 0
Example 23:
x4 – 5x2 – 36 = 0
Reflection:
15
Sample CCSD Common Exam Practice Question(s):
1. Which of the following represents the solution set of the polynomial equation
x 4  7 x 2  12  0 ?
A.
B.
C.
D.
2, 2, i 3, i 3
2, 2, 3,  3
2i, 2i, i 3, i 3
2i, 2i, 3,  3
2. What is the factored form of the polynomial x 3  27 ?
A.
 x  3  x 2  3 x  9 
B.
 x  3  x 2  3 x  9 
C.
 x  3  x 2  3 x  9 
D.
 x  3  x 2  3 x  9 
Reflection:
16