5.2: Evaluate and Graph Polynomial Functions Objectives: Assignment: To evaluate

5.2: Evaluate and Graph Polynomial Functions
Objectives:
1. To evaluate
polynomial functions
using direct
substitution and
synthetic substitution
2. To describe the end
behavior of the
graphs of polynomial
functions
Assignment:
• P. 341-344: 1, 2-22
even, 24-37, 51, 54,
61, 62
• Secant Line
Worksheet
Warm-Up 1
How does the sign of a in f (x) = ax2 + bx + c
affect the graph of f (x)?
y
y
a>0
x
a<0
x
Warm-Up 2
Use synthetic division to divide 3𝑥 3 + 5𝑥 2 − 7𝑥 − 4
by 𝑥 + 2
Vocabulary
Polynomial
Coefficient
Leading Coefficient
Constant Term
Degree
Linear
Quadratic
Cubic
Quartic
Synthetic Division
Objective 1
You will be able to
evaluate polynomial
functions using direct
substitution and synthetic
substitution
Polynomial
A polynomial in x is an expression of the
form:
n
2
an x 
 a2 x  a1 x  a0
Exponents:
All exponents
are
nonnegative
integers
The degree
of the
polynomial is
n, as long as
an  0
The degree
is the highest
power of the
polynomial
Polynomial
A polynomial in x is an expression of the
form:
n
2
an x 
 a2 x  a1 x  a0
Coefficients:
All
coefficients ak
are real
numbers
an is the
leading
coefficient
a0 is the
constant term
Polynomial
A polynomial in x is an expression of the
form:
n
2
an x 
 a2 x  a1 x  a0
Standard form:
Variables written with powers in descending
order
Types and Degrees of Polys
Degree
Type
Standard Form
0
Constant
f(x) = a0
1
Linear
f(x) = a1x + a0
2
Quadratic
f(x) = a2x2 + a1x + a0
3
Cubic
f(x) = a3x3 + a2x2 + a1x + a0
4
Quartic
f(x) = a4x4 + a3x3 + a2x2 + a1x + a0
5
Quintic
f(x) = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
6
Sextic
f(x) = a6x6 + a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
Types and Degrees of Polys
Degree
Type
Examples
0
Constant
f(x) = 42
1
Linear
f(x) = 3x – 7
2
Quadratic
f(x) = -1.5x2 + .75x + 6
3
Cubic
f(x) = x3 + 4x2 – 5x + 1
4
Quartic
f(x) = 7x4 + 9.5x3 +2x2 + 10x – 1
5
Quintic
f(x) = 4x5 + 8x4 + 15x3 + 16x2 + 23x + 42
6
Sextic
f(x) = x6 – 2x5 + 3x4 – 4x3 + 5x2 – 6x + 7
Evaluating Functions
Evaluating a function at a particular value
simply means finding the output for a given
input. This is commonly (i.e., tediously)
done with direct substitution.
1. Input your chosen value for every x.
2. (Arithmetic)
3. Your answer is the output value, f(x).
Exercise 1
Use direct substitution to evaluate the
function below at each of the follow values.
f ( x)  x 3  5 x 2  6 x  1
1. x = -2
2. x = 4
Exercise 2
Use synthetic division to divide the function
below by each of the following monomials.
f ( x)  x 3  5 x 2  6 x  1
1. x + 2
2. x – 4
Remainder Theorem
If a polynomial f(x) is divided by x – k, then
the remainder is f(k).
When using synthetic division to evaluate a
function, it is called synthetic substitution.
This means that we can
use synthetic division to
quickly evaluate a function
at a particular x-value.
The important thing to
remember here is that
you do not change the
sign of k!
Exercise 3
Use synthetic substitution to evaluate the
polynomial function for the given value of x.
1.
f ( x)  5x3  3x2  x  7; x  2
2.
g ( x)  2 x4  x3  4 x  5; x  1
Exercise 4
What does it mean if the remainder is zero
when you perform synthetic substitution?
Objective 2
You will be able
to describe the
end behavior of
polynomial
graphs
Activity: End Behavior
In this activity, we will
investigate the look,
feel, and taste of the
“end” behavior of
polynomial graphs.
(“End” is in quotation
marks because, strictly
speaking, most graphs
don’t have “ends”.)
Activity: End Behavior
End behavior refers to
what a graph looks like
as x approaches positive
infinity
x  
or as x approaches
negative infinity
x  
End Behavior: Linear
End Behavior: Quadratic
End Behavior: Cubic
End Behavior: Quartic
End Behavior: Quintic
End Behavior: Even and Odd
Now look at all the functions of even degree. What
do you notice about their end behavior? What
about the end behavior of odd functions?
End Behavior: Even and Odd
Even End Behavior
Odd End Behavior
Exercise 5
Describe the degree
and leading
coefficient of the
polynomial whose
graph is shown.
Exercise 6a
Let f (x) = 35x26 − 2x15 + 5. Describe the end
behavior of f (x).
Exercise 6b
Let g(x) = −35x37 + 13x20 − 7. Describe the
end behavior of g(x).
Exercise 7
Let f (x) be represented by the graph below.
If g(x) = −x ∙ f (x), then describe the end
behavior of g(x).
y
x
5.2: Evaluate and Graph Polynomial Functions
Objectives:
1. To evaluate
polynomial functions
using direct
substitution and
synthetic substitution
2. To describe the end
behavior of the
graphs of polynomial
functions
Assignment
• P. 341-344: 1, 2-22
even, 24-37, 51, 54,
61, 62
• Secant Line
Worksheet
“Daddy, do you think my end behavior is odd?”