SECTION 4.1POLYNOMIAL FUNCTIONS
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WHAT IS A POLYNOMIAL FUNCTION?
A function that contains powers of “x” in the form of:
A,B,C,D,…E are Coefficients & must be real numbers
The exponents “n”, “n-1”…. have to be whole numbers
Can’t have “x” in the denominator or “x” as a radical
Ex#1) Indicate which of the following are not polynomials:
COMMON TERMS FOR POLYNOMIAL:
Degree of a Polynomial:
The largest exponent for the variable
Leading Coefficient:
The coefficient of the term with the largestexponent
COMMON POLYNOMIAL FUNCTIONS:
GENERAL CONCEPTS:
EVEN Degree functions
opens up
ODD Degree functionsusually begin at the leftside from the bottom
If the leading coefficientis NEGATIVE, thegraph will have avertical reflection (Upside down)
EX: GIVEN THE FOLLOWING GRAPH, WHICH OF THE FOLLOWINGEQUATIONS IS THE CORRECT ONE?
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10
-5
5
10
EX: GIVEN THE FOLLOWING GRAPH, WHICH OF THEFOLLOWING EQUATIONS IS THE CORRECT ONE?
TERMS:
Absolute Max: the point on a function with the greatest(y-coordinate)
Absolute Minimum: the point on a function with thesmallest (y-coordinate)
Relative Max: (Local Max) A maximum point on top of a“hill”
Relative Min.: (Local Min) A minimum point on thebottom of a “valley”
If a graph extends up to positive infinity, there will not beany “absolute Max”
If a graph extends down to negative infinity, there willnot be any “absolute Min”
EX: FIND THE ABS/REL. MAX & MIN
A
B
C
D
EX: FIND ALL THE RELATIVE/ABSOLUTEMAX & MIN.
A
B
C
D
E
A
C
E
B
D
A
II) FINDING MAX & MIN:
Ex: Given the function:
a)Sketch a graph
b)Determine any Maximum or Minimum
c)Find the Domain and Range
Use Ti-83: Graph thefunction
Ex: Graph
Find any Max or Min
Find the Domain and Range
III) READING A FUNCTION IN FACTORED FORM:
A polynomial function in General form:
A polynomial function in Factored Form:
EX: DETERMINE THE EQUATION OF A FUNCTION WITHROOTS: -1, 1, 3 AND Y-INTERCEPT OF -6.
Plug the roots into the equation in factored form
Plug the coordinates of the point into the equation tosolve for “a”
IV) HOW TO GRAPH A POLYNOMIALFUNCTION IN FACTORED FORM:
Find the roots (x-intercept)look for repeated roots
Find the Y-intercept,where the graph crosses the y-axis
Find the DegreeCount the number of X’s in the function
Find the lead coefficientWhich way the graph opens
EX: GIVEN THE EQUATION:GRAPH AND FIND THE FOLLOWING
EX: GIVEN THE EQUATION:FIND THE FOLLOWING:
IV) REPEATED ROOTS:
When 2 roots are the same, it’s called a “Double Root”.
Instead of crossing the x-axis, it “bounces” off the x-axis.
When 3 roots are the same, it’s called a “Triple Root”
EX: GIVEN THE EQUATION:GRAPH AND FIND THE FOLLOWING
x
y
-2
-1
0
1
-15
-10
-5
EX: GIVEN THE EQUATION:GRAPH AND FIND THE FOLLOWING
