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6.3
Evaluating and Graphing Polynomial Functions
A polynomial function is a function of the form
f (x) = an x n + an – 1 x
n – 1 +· · ·+ a 1 x + a 0
Where an 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
an 0
an
an leading coefficient
a 0
a0 constant term n
n
degree
descending order of exponents from left to right.
n n – 1
Degree Type Standard Form
You are already familiar with some types of polynomialfunctions. Here is a summary of common types ofpolynomial functions.
4 Quartic f (x) = a4 x 4 + a 3 x
3 + a 2 x 2 + a 1 x + a
0 Constant f (x) = a
3 Cubic f (x) = a 3 x 3 + a 2 x
2 + a 1 x + a
2 Quadratic f (x) = a 2 x 2 + a 1 x + a
1 Linear f (x) = a1x + a
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
f (x) = x 2
– 3x4 – 712
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.
Its standard form is f (x) = – 3x 4
+ x 2 – 7. 1
2
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
Identifying Polynomial Functions
The function is not a polynomial function because the
term 3
x does not have a variable base and an exponentthat is a whole number.
SOLUTION
f (x) = x 3 + 3
x
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
SOLUTION
f (x) = 6x 2 + 2 x
–1 + x
The function is not a polynomial function because the term2x
–1 has an exponent that is not a whole number.
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 2, so it is a quadratic function.
The leading coefficient is .
Its standard form is f (x) = x2 – 0.5x – 2.
f (x) = – 0.5 x + x 2 – 2
f (x) = x 2 – 3 x
4 – 712
Identifying Polynomial Functions
f (x) = x 3 + 3x
f (x) = 6x2 + 2 x– 1 + x
Polynomial function?
f (x) = – 0.5x + x2 – 2
Using Synthetic Substitution
One way to evaluate polynomial functions is to usedirect substitution.
Use substitution to evaluateUse substitution to evaluate
f (x) = 2 x 4 + 8 x
2 + 5 x 7 when x = 3.
Now use direct substitution:
f (x) = 2 x 4 + 8 x
2 + 5 x 7 when x = 3.
7)3(5)3(8)3(2)( 24 xf
98
Using Synthetic Substitution
One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.
Use synthetic division to evaluate
f (x) = 2 x 4 + 8 x
2 + 5 x 7 when x = 3.
Polynomial in standard form
Using Synthetic Substitution
2 x 4 + 0 x
3 + (–8 x 2) + 5 x + (–7)
2 6
6
10
18
35
30 105
98
The value of f (3) is the last number you write,In the bottom right-hand corner.
The value of f (3) is the last number you write,In the bottom right-hand corner.
2 0 –8 5 –7 CoefficientsCoefficients
3
x-value
3 •
SOLUTION
Polynomial instandard form
Start by writing it in standard form
Using Synthetic Substitution
Use synthetic division to evaluate
f (x) = x 2 - x
5 + 1 x = -1.
Using Synthetic Substitution
Use synthetic division to evaluate
f (x) = x 2 - x
5 + 1 x = -1.
Did you get 3?
HOMEWORK (DAY 1)
Pg # 333 #16 – 26, 30- 34, 38-46
(Evens only)
If “n” is even, the graph of the polynomial is “U-shaped” meaning it is parabolic (the higher the degree, the more curves the graph will have in it).
If “n” is odd, the graph of the polynomial is “snake-like” meaning looks like a snake (the higher the degree, the more curves the graph will have in it).
Let’s talk about the Leading Coefficient Test:
Leading Coefficient Test
Degree is evenDegree is odd
Leading coefficient is positive
Start high,End high
Leading coefficient is negative
Start low,End low
Leading coefficient is positive
Start low,End high
Leading coefficient is negative
Start high,End low
with a positiveleading coefficient
with a negativeleading coefficient
with a positiveleading coefficient
with a negativeleading coefficient
Determine the left and right behavior of the graph of each polynomial function.
f(x) = -x5 +3x4 – x
f(x) = x4 + 2x2 – 3x
f(x) = 2x3 – 3x2 + 5
Determine the left and right behavior of the graph of each polynomial function.
f(x) = -x5 +3x4 – x
f(x) = x4 + 2x2 – 3x
f(x) = 2x3 – 3x2 + 5
Even, Leading coefficient 1 (positive) , starts high ends high
Odd, Leading coefficient 1 (negative) , starts high ends low
ODD , Leading coefficient 2 (positive) , starts LOW ends HIGH
Tell me what you know about the equation…
Odd / Even ?
Leading coefficient Positive or Negative?
Tell me what you know about the equation…
Odd / Even ?
Leading coefficient Positive or Negative?
Tell me what you know about the equation…
Odd / Even ?
Leading coefficient Positive or Negative?
Tell me what you know about the equation…
Fundamental Thm of AlgebraZeros of Polynomial Functions:1. The graph of f has at most n
real zeros
The “n” deals with the highest exponent!
How many zeros do these graphs have????
GRAPHING POLYNOMIAL FUNCTIONS
The end behavior of a polynomial function’s graphis the behavior of the graph as x approaches infinity(+ ) or negative infinity (– ). The expressionx + is read as “x approaches positive infinity.”
GRAPHING POLYNOMIAL FUNCTIONS
END BEHAVIOR
x
f(x)–3 –2 –1 0 1 2 3
Graphing Polynomial Functions
Graph f (x) = x 3 + x
2 – 4 x – 1.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
x
f(x)–3
–7
–2
3
–1
3
0
–1
1
–3
2
3
3
23
Graphing Polynomial Functions
Graph f (x) = x 3 + x
2 – 4 x – 1.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
x
f (x)
–3 –2 –1 0 1 2 3
Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x
3 + 2x 2 + 4x.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
x
f (x)
–3
–21
–2
0
–1
–1
0
0
1
3
2
–16
3
–105
Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x
3 + 2x 2 + 4x.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
Assignment
Pg 334 # 49 – 52, 54- 64, 66-72 evens