POLYNOMIALS REVIEW
The DEGREE of a polynomial is the largest
degree of any single term in the polynomial
(Polynomials are often written in descending
order of the degree of its terms)
COEFFICIENTS are the numerical value of each
term in the polynomial
The LEADING COEFFICIENT is the numerical
value of the term with the HIGHEST DEGREE.
Polynomials Review Practice
For each polynomial
1) Write the polynomial in descending order
2) Identify the DEGREE and LEADING COEFFICIENT
of the polynomial
7
5
9
3
11 x  8 x  2 x  15 x  5 x
 x  7x  9x  6x  1
8
6
4
2
3x  4x  5x  2x  7
2
4
3
4 x 5  6 x 4  3 x 2  11 x  5
Finding values of a polynomial:
Substitute values of x into polynomial and simplify:
f ( x )  4 x  6 x  3 x  11 x  5
5
4
2
f ( 2)  __________
f (2)  4(2)  6(2)  3(2)  11(2)  5  185
5
4
2
Find each value for f ( x )  x  5 x  12 x  6
1. f ( 3)
2. f (1)
3
1
3. f ( 2 )
2
4. f ( 3a 2 )
Graphs of Polynomial Functions:
Constant Function
(degree = 0)
Linear Function
(degree = 1)
Cubic Function
(deg. = 3)
Quartic Function
(deg. = 4)
Quadratic Function
(degree = 2)
Quintic Function
(deg. = 5)
OBSERVATIONS of Polynomial Graphs:
1) How does the degree of a polynomial function relate the
number of roots of the graph?
The degree is the maximum number of zeros or roots that a
graph can have.
2) Is there any relationship between the degree of the
polynomial function and the shape of the graph?
Number of Changes in DIRECTION OF THE GRAPH = DEGREE
EVEN DEGREES: Start and End both going UP or DOWN
ODD DEGREES: Start and End as opposites  UP and DOWM
OBSERVATIONS of Polynomial Graphs:
3) What additional information (value) related the degree of
the polynomial may affect the shape of its graph?
LEADING COEFFICIENT
Numerical Value of Degree
ODD DEGREE:
EVEN DEGREE:
POSITIVE Leading Coefficient POSITIVE Leading Coefficient
= START Down and END Up
= UP
NEGATIVE Leading Coefficient NEGATIVE Leading Coefficient
= START Up and END Down
= DOWN
Describe possible shape of the following based on the
degree and leading coefficient:
f ( x)  2 x  3 x  5
4
2
g( x )  3 x  7 x  4 x  1
5
3
Degree Practice with Polynomial Functions
• Identify the degree as odd or even and state the assumed degree.
• Identify leading coefficient as positive or negative.
Draw a graph for each descriptions:
Description #1:
Degree = 4
Leading Coefficient = 2
Description
#4:
Degree = 8
Leading
Coefficient
= -2
Description #2:
Degree = 6
Leading Coefficient = -3
Description #3:
Degree = 3
Leading Coefficient = 1
Description
#5:
Degree = 5
Leading
Coefficient
= -4
Graphs # 1 – 6 Identify RANGE: Interval or Inequality Notation
Graph #3
Graph #2
Graph #1
(-2, 8)
(1, 4)
(0, 11)
(13, 9)
(7, -2)
(-17, -10)
(-6, -9)
(-5, -9)
Range, y: (-∞,
∞)
(4, -15)
Range, y: (-15,
Graph #5
Graph #4
(-3,12)
(-3, 3)
∞)
(6, 11)
Range, y,: (-∞,
∞)
(-5,17) Graph #6
(1, 12)
(4, 8)
(2, 2)
(-2, 6)
(3, 2)
(-5, -4)
(1, -3)
(4, -5)
Range, y: (-5,
∞)
(1, -9)
Range, y: (-∞,
12 )
Range, y,: (-∞,
17 )
The END
BEHAVIOR of a polynomial describes the
RANGE, f(x), as the DOMAIN, x, moves
LEFT (as x approaches negative infinity: x → - ∞) and
RIGHT (as x approaches positive infinity : x → ∞)
on the graph.
Determine the end behavior for each of the given graphs
Increasing
to the Left
Decreasing
to the
Left
Decreasing
to the
Right
Decreasing
to the
Right
Right: x   f ( x )  
Right: x   f ( x )  
Left: x   f ( x )  
Left: x  
f ( x)  
Use Graphs #1 – 6 from the previous Slide
• Describe the END BEHAVIOR of each graph
• Identify if the degree is EVEN or ODD for the graph
• Identify if the leading coefficient is POSITIVE or NEGATIVE
GRAPH #3
GRAPH #2
GRAPH #1
x  , f ( x )   x  , f ( x )   x  , f ( x )  
x  , f ( x )   x  , f ( x )   x  , f ( x )  
Degree: ODD
LC: NEG
Degree: EVEN
LC: POS
Degree: ODD
LC: NEG
GRAPH #6
GRAPH #5
x  , f ( x )   x  , f ( x )  
x  , f ( x )   x  , f ( x )   x  , f ( x )  
GRAPH #4
x  , f ( x )  
Degree: EVEN
LC: POS
Degree: EVEN
LC: NEG
Degree: EVEN
LC: NEG
Describing Polynomial Graphs Based on the Equation
Based on the given polynomial function:
• Identify the Leading Coefficient and Degree.
• Sketch possible graph (Hint: How many direction changes possible?)
• Identify the END BEHAVIOR
f ( x)  2 x5  6 x 3  3 x
Degree: 5  Odd
LC: 2  Pos
Start Down, End Up
x  , f ( x )  
x  , f ( x )  
4
Degree: 6  Even
LC: 1  Pos
Start Up, End Up
x  , g ( x )  
x  ,
g( x )  
Degree: 4  Even
LC: -1  Neg
Start Down, End Down
x  , h( x )  
x  , h( x ) 
g( x )  x  2 x  3 x  4
6
h( x )   x 4  x 2  2 x  1
2

p( x )  2 x 3  x 2  3 x  x  3
Degree: 3  Odd
LC: -2  Neg
Start Down, End Up
x  , p( x )  
x  ,
p( x )  
EXTREMA: MAXIMUM and MINIMUM
points are the highest and lowest points on the graph.
• Point A is a Relative Maximum because it
is the highest point in the immediate area
(not the highest point on the entire graph).
• Point B is a Relative Minimum because it
is the lowest point in the immediate area
(not the lowest point on the entire graph).
• Point C is the Absolute Maximum
because it is the highest point on the entire
graph.
• There is no Absolute Minimum on this
graph because the end behavior is:
x  , f ( x )  
x  , f ( x )  
(there is no bottom point)
C
A
B
Identify ALL Maximum or Minimum Points
Distinguish if each is RELATIVE or ABSOLUTE
Graph #1
Graph #2
R: Max
R: Max (-2, 8)
(1, 4)
Graph #3 R: Max
(0, 11)
(13, 9)
R: Max
(7, -2)
(-6, -9)
(-5, -9)
(4, -15)
R: Min
R: Min
A: Min
Graph #5
Graph #4
R: Max
(-3, 3) R: Max
(-3,12)
(6, 11)
A: Max
(2, 2)
R: Min
(-17, -10)
R: Min
R: Max
Graph #6
(-2, 22)
R: Max
(6, 3)
(-5, -4)
(1, -3)
(4, -5)
(1, -9)
R: Min
R: Min
A: Min
R: Min
R: Min
CALCULATOR COMMANDS for
POLYNOMIAL FUNCTIONS
The WINDOW needs to be large enough to see graph!
• The ZEROES/ ROOTS of a polynomial function are
the x-intercepts of the graph.
Input [ Y=] as Y1 = function and Y2 = 0
[2nd ] [Calc]  [Intersect]
• To find EXTEREMA (maximums and minimums):
Input [ Y=] as Y1 = function
[2nd ][Calc]  [3: Min] or [4: Max]
– LEFT and RIGHT bound tells the calculator where on the
domain to search for the min or max.
– y-value of the point is the min/max value.
Using your calculator: GRAPH the each
polynomial function and IDENTIFY the ZEROES,
EXTREMA, and END BEHAVIOR.
4
3
f
(
x
)


x

3
x
 8 x  11 [2] g( x )  2 x  6 x  4 x  1
[1]
3
2
1 5
[3] y  x  x  8 x  5 x  1 [4] y   x  3 x 3  2 x 2  8 x
2
4
3
2