Warm Up
The senator chose to incur
dislike rather than ------- her
principles to win favor with the
public.
• (A) gratify (B) endorse
• (C) accuse (D) compromise
• (E) advertise
Algebra 3
Chapter 6
Polynomials and Polynomial
Functions
Lesson 2 Evaluating and Graphing
Polynomials
VOCAB
• Function – is another name of an equation…instead of y you
will see f(x)
• Polynomial function – is a function in the form. (formula on
board)
• Leading coefficient – the number in front of the biggest
power
• Degree – is the highest power of the function
• Constant term – is the number (it has no x)
• Standard Form– is when the terms are written in decreasing
order of exponents
• Synthetic Substitution – is a cool easy way to solve
polynomial functions
• End behavior– is telling what the graph is doing as x
approaches infinity and as x approaches negative infinity
Directions (ID polynomials)
• Write the function in standard form
Determining Polynomials
• Polynomial exponents must be a positive
whole number
I DO ( ID polynomials)
• Is it a polynomial?
– If yes…what is it’s degree and leading coefficient
1. 𝑓 𝑥 =
1 2
𝑥
2
3
− 3𝑥 4 − 7
2. 𝑓 𝑥 = 𝑥 + 3𝑥
3. 𝑓 𝑥 = 6𝑥 2 + 2𝑥 −1 + 𝑥
4. 𝑓 𝑥 = −.5𝑥 + 𝜋𝑥 2 − 2
WE DO ( ID polynomials)
• Is it a polynomial?
– If yes…what is it’s degree and leading coefficient
1. 𝑓 𝑥
2. 𝑓 𝑥
3. 𝑓 𝑥
4. 𝑓 𝑥
= 3𝑥 2 + 7𝑥 − 3
=9
= 2 𝑥 + 5𝑥 − 8
= 5 − 3𝑥 4
YOU DO ( ID polynomials)
• Is it a polynomial?
– If yes…what is it’s degree and leading coefficient
1. 𝑓 𝑥 = 2𝑥 − 3𝑥 + 1
2. 𝑓 𝑥 = 𝑥 3 + 𝑥 2 − 𝜋
3. 𝑓 𝑥 = 3𝑥 2 − 5 + 2𝑥 3
4. 𝑓 𝑥 = 𝑥 + 3 − 5𝑥 3 + 7𝑥 2
Review
• Today you learned how to identify a
polynomial function
Homework
• 6.2B (1 – 6)
Warm Up
• Is it a polynomial?
– If yes…what is it’s degree and leading coefficient
1. 𝑓 𝑥 = 3
2. 𝑓 𝑥 = 2 𝑥 + 𝑥 7 − 𝜋
3. 𝑓 𝑥 = 3𝑥 3 − 5 + 2𝑥 5
4. 𝑓 𝑥 = 𝑥 + 3 − 51𝑥 + 7𝑥 2
Algebra 3
Chapter 6
Polynomials and Polynomial
Functions
Lesson 2 Evaluating and Graphing
Polynomials
Directions (Evaluation)
• Plug x in and solve
I DO (Evaluating)
Use direct substitution to evaluate the function
for x
1. 𝑓 𝑥 = 3 − 𝑥 2 + 4𝑥 − 𝑥 3
for x = 2
2. 𝑓 𝑥 = 7𝑥 + 2𝑥 2 − 5
for x = 3
3. 𝑓 𝑥 = 3𝑥 2 + 5𝑥 − 2𝑥 5 + 𝑥 4
for x = -1
4. 𝑓 𝑥 = −𝑥 2 + 5𝑥 + 22
for x = -4
WE DO (Evaluating)
Use direct substitution to evaluate the function
for x
1. 𝑓 𝑥 = 3𝑥 3 − 4𝑥 2 + 𝑥 − 7 for x = 2
2. 𝑓 𝑥 =
3 3
1 2
𝑥 − 𝑥 + 3𝑥
2
4
4
2
− 1 for x = 2
3. 𝑓 𝑥 = −𝑥 + 2𝑥 + 5
for x = − 5
4. 𝑓 𝑥 = 2𝑥 3 + 5𝑥 2 + 4𝑥 + 8 for x= -2
YOU DO (Evaluating)
Use direct substitution to evaluate the function
for x
1. 𝑓 𝑥 = 𝑥 +
1 3
𝑥
2
for x = 4
2. 𝑓 𝑥 = 5𝑥 4 − 8𝑥 3 + 7𝑥 2
3. 𝑓 𝑥 = 3𝑥 2 − 5 + 2𝑥 3
4. 𝑓 𝑥 = 𝑥 + 3 − 5𝑥 3 + 7𝑥 2
for x = 1
for x = 0
for x = -3
Review
• What did you learn today?
• Tomorrow
– Synthetic Substitution
Homework
• Worksheet
– 6.2B (7 – 14)
Warm Up
A right circular cylinder has
height of 6 and volume of
54𝜋. What is the
circumference of its base?
A.2 𝜋 B. 3 𝜋 C. 6 𝜋
D. 9 𝜋 E. 18 𝜋
Algebra 3
Chapter 6
Polynomials and Polynomial
Functions
Lesson 2 Evaluating and Graphing
Polynomials
Directions (Synthetic Substitution)
• Write the problem in standard form
• Write down the number you want to plug in
• Write down the coefficient to ALL terms (even the
ones that you don’t see)
• Drop the first term down
• Multiply by the 1st number
• Add going down
I DO (Synthetic Substitution)
Use synthetic substitution to evaluate
• 1. 𝑓 𝑥 = 2𝑥 4 − 8𝑥 2 + 5𝑥 − 7
when x = 3
• 2. 𝑓 𝑥 = 5𝑥 3 + 4𝑥 2 + 8𝑥 + 1
when x = 2
• 3. 𝑓 𝑥 = 𝑥 3 + 3𝑥 2 + 6𝑥 − 11
when x = -5
3
• 4. 𝑓 𝑥 = −4𝑥 + 3𝑥 − 5
when x =
1
2
WE DO (Synthetic Substitution)
Use synthetic substitution to evaluate
• 1. 𝑓 𝑥 = 2𝑥 4 + 𝑥 3 − 3𝑥 2 + 5𝑥
• 2. 𝑓 𝑥 = 2𝑥 3 − 𝑥 2 + 6𝑥
• 3. 𝑓 𝑥 = −3𝑥 3 + 7𝑥 2 − 4𝑥 + 8
• 4. 𝑓 𝑥 = 𝑥 3 − 𝑥 2 + 12𝑥 + 15
when x = -1
when x = 5
when x = 3
when x = -1
YOU DO (Synthetic Substitution)
Use synthetic substitution to evaluate
• 1. 𝑓 𝑥 = −𝑥 4 + 𝑥 3 − 𝑥 + 1
• 2. 𝑓 𝑥 = 3𝑥 5 − 2𝑥 2 + 𝑥
• 3. 𝑓 𝑥 = −𝑥 4 + 8𝑥 3 + 13𝑥 − 4
• 4. 𝑓 𝑥 = 3𝑥 4 − 2𝑥 2 + 5
when x = -3
when x = 2
when x = -2
when x = 2
Review
• Today we learned how to use synthetic
substitution to solve a polynomial
Homework
• Worksheet
– 6.2B (15 – 22)
Warm Up
Use synthetic substitution to
evaluate
5
2
1. 𝑓 𝑥 = −𝑥 + 2𝑥 − 𝑥 + 7
when x = 7
Algebra 3
Chapter 6
Polynomials and Polynomial
Functions
Lesson 2 Evaluating and Graphing
Polynomials
Directions (Graphing Knowledge)
• Write the problem in standard form
• Look at the Degree And Leading Coefficient
– Today is all about graphs behavior
– Tomorrow we will see the graphs
𝒇(𝒙) as x
APPROACHES
−∞
𝒇(𝒙) as x
APPROACHES
∞
Degree
LC
SHAPE
EVEN
POSITIVE
U
∞
∞
EVEN
NEGATIVE
n
−∞
−∞
ODD
POSITIVE
On the board
−∞
∞
ODD
NEGATIVE
On the board
∞
−∞
Exceptions to the rule
• There are only 2 exceptions
– Linear Equations
• Talked about in Algebra 1
Degree
LC
Shape
𝒇(𝒙) as x
APPROACHES
−∞
𝒇(𝒙) as x
APPROACHES
∞
0
Any
Horizontal Line
LC
LC
1
Positive
See Board
−∞
∞
1
Negative
See Board
∞
−∞
I DO (Graphing Knowledge)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 2𝑥 8 − 8𝑥 12 + 5𝑥 − 7
• 2. 𝑓 𝑥 = 5𝑥 3 + 4𝑥 7 + 8 + 𝑥
• 3. 𝑓 𝑥 = −𝑥 15 − 3𝑥 24 + 6𝑥 − 11
• 4. 𝑓 𝑥 = 4𝑥 8 + 3𝑥 − 5
WE DO (Graphing Knowledge)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 7𝑥 13 + 𝑥 47 − 𝑥 + 7
• 2. 𝑓 𝑥 = 2𝑥 2
• 3. 𝑓 𝑥 = −𝑥 29 − 3𝑥 31 + 6𝑥
• 4. 𝑓 𝑥 = −4𝑥 6 + 3𝑥 − 5
YOU DO (Graphing Knowledge)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 3𝑥 2 − 8𝑥 6 + 7𝑥 + 13
• 2. 𝑓 𝑥 = −5𝑥 3
• 3. 𝑓 𝑥 = 𝑥 6 + 3𝑥 9 + 19
• 4. 𝑓 𝑥 = 4𝑥 1024 + 𝑥 − 92
Review
• Today we learned how to determine what a
polynomial graph looks like without seeing the
graph
Homework
• Worksheet
– 6.2B (23 - 31)
Warm Up
Use graphing knowledge
5
2
1. 𝑓 𝑥 = −𝑥 + 2𝑥 − 𝑥 + 7
Algebra 3
Chapter 6
Polynomials and Polynomial
Functions
Lesson 2 Evaluating and Graphing
Polynomials
Directions (Graphing)
• Type it into the calculator
– Remember ^ is how you put big power
• Draw the graph
I DO (Graphing)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 2𝑥 8 − 8𝑥 12 + 5𝑥 + 3
• 2. 𝑓 𝑥 = 5𝑥 3 + 4𝑥 7 + 8 + 𝑥
• 3. 𝑓 𝑥 = −𝑥 15 − 3𝑥 24 + 6𝑥 − 1
• 4. 𝑓 𝑥 = 4𝑥 8 + 3𝑥 − 5
WE DO (Graphing)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 7𝑥 13 + 𝑥 47 − 𝑥 + 2
• 2. 𝑓 𝑥 = 2𝑥 2
• 3. 𝑓 𝑥 = −𝑥 29 − 3𝑥 31 + 6𝑥
• 4. 𝑓 𝑥 = −4𝑥 6 + 3𝑥 + 5
YOU DO (Graphing)
Use graphing knowledge to tell me about the graph
• 1. 𝑓 𝑥 = 3𝑥 2 − 8𝑥 6 + 7𝑥 + 1
• 2. 𝑓 𝑥 = −5𝑥 3
• 3. 𝑓 𝑥 = 𝑥 6 + 3𝑥 9 − 3
• 4. 𝑓 𝑥 = 4𝑥 1024 + 𝑥 − 2
Review
• Today we learned how to draw a polynomial
graph.
Homework
• Worksheet
– 6.2B (23 - 31)