POLYNOMIALS REVIEW

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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.
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• Polynomials Review Practice
• For each polynomial
• Write the polynomial in descending order
• Identify the DEGREE and LEADING COEFFICIENT of
  the polynomial

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Finding values of a polynomial Substitute
values of x into polynomial and simplify
Find each value for 1. 2.
3. 4.
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Graphs of Polynomial Functions
              Constant Function Linear
Function Quadratic Function (degree
0) (degree 1) (degree
2)               Cubic Function Quartic
Function Quintic Function (deg.
3) (deg. 4) (deg. 5)
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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
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Describe possible shape of the following based on
the degree and leading coefficient
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 POSITIVE Leading Coefficient
START Down and END Up NEGATIVE Leading
Coefficient START Up and END Down
EVEN DEGREE POSITIVE Leading Coefficient
UP NEGATIVE Leading Coefficient DOWN
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Degree Practice with Polynomial Functions

• Identify the degree as odd or even and state the
  assumed degree.
• Identify leading coefficient as positive or
  negative.

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Draw a graph for each descriptions
Description 1 Degree 4 Leading Coefficient 2
Description 2 Degree 6 Leading Coefficient
-3
Description 3 Degree 3 Leading Coefficient 1
Description 5 Degree 5 Leading Coefficient
-4
Description 4 Degree 8 Leading Coefficient
-2
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Graphs 1 6 Identify RANGE Interval or
Inequality Notation
Graph 3
Graph 2
Graph 1
(-2, 8)
(0, 11)
(13, 9)
(1, 4)
(7, -2)
(-17, -10)
(-6, -9)
(-5, -9)
(4, -15)
Range, y (-8, 8 )
Range, y, (-8, 8 )
Range, y (-15, 8 )
Graph 6
Graph 5
(-5,17)
Graph 4
(-3,12)
(6, 11)
(1, 12)
(-3, 3)
(4, 8)
(2, 2)
(-2, 6)
(3, 2)
(1, -3)
(-5, -4)
(1, -9)
(4, -5)
Range, y (-8, 12 )
Range, y, (-8, 17 )
Range, y (-5, 8 )
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The END BEHAVIOR of a polynomial describes the
RANGE, f(x), as the DOMAIN, x, moves LEFT (as x
approaches negative infinity x ? - 8) and RIGHT
(as x approaches positive infinity x ? 8) 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 Left
Right Left
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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
Degree ODD LC NEG
Degree EVEN LC POS
Degree ODD LC NEG
GRAPH 6
GRAPH 5
GRAPH 4
Degree EVEN LC NEG
Degree EVEN LC POS
Degree EVEN LC NEG
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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

Degree 4 ? Even LC -1 ? Neg Start Down, End Down
Degree 5 ? Odd LC 2 ? Pos Start Down, End Up
Degree 3 ? Odd LC -2 ? Neg Start Down, End Up
Degree 6 ? Even LC 1 ? Pos Start Up, End Up
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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
• (there is no bottom point)

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Identify ALL Maximum or Minimum
PointsDistinguish if each is RELATIVE or ABSOLUTE
Graph 3
Graph 2
R Max
R Max
R Max
(-2, 8)
(0, 11)
(13, 9)
R Max
(7, -2)
(-6, -9)
R Min
(-17, -10)
(4, -15)
R Min
R Min
R Min
A Min
Graph 6
Graph 5
Graph 4
R Max
R Max
(-3,12)
(6, 11)
(-2, 22)
R Max
(-3, 3)
A Max
(2, 2)
R Max
(6, 3)
R Min
(1, -3)
(1, -9)
(-5, -4)
(4, -5)
R Min
R Min
R Min
A Min
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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.

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Using your calculator GRAPH the each polynomial
function and IDENTIFY the ZEROES, EXTREMA, and
END BEHAVIOR.
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