7.1 polynomial functions objective: evaluate polynomials identify general shapes of graphs of...
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7.1 Polynomial Functions7.1 Polynomial Functions
•Objective: •Evaluate Polynomials•Identify general shapes of graphs of polynomials
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Polynomial FunctionPolynomial Function(in one variable)(in one variable)
degree of f(x): highest exponent
leading coefficient: the coefficient of the term with the highest degree
1 20 1 2 1( ) ...n n
n n nf x a x a x a x a x a
0a
0 0a
n
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Examples:Examples:State the degree & leading
coefficient
1.
2.
3.
3 27 4z z z
3 2 26 4a a ab
2 49 3y y y
degree: 3
leading coeff.: 7
not a poly in 1 variable
degree: 4
leading coeff.: 1
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Evaluate a Polynomial Evaluate a Polynomial FunctionFunction
Find f(6)
Find g(3)
2( ) 3 3 1f r r r 2(6) 3(6) 3(6) 1 37f
3( ) 2 4 ( )g x x x 3(3) 2 4(3) ( (3) )g
2 12 ( 27) 41
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Find f(a2)
Find f(h+3)
2( ) 3 3 1f r r r
2 2 2 2( ) 3( ) 3( ) 1f a a a 4 23 3 1a a
2( 3) 3( 3) 3( 3) 1f h h h 23( 6 9) 3 9 1h h h
23 15 19h h
23 18 27 3 8h h h
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Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsConstant
Functiondegree = 0
=4
Linear Functiondegree = 1
4y 2 1y x
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Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsQuadratic
Functiondegree = 2
=4
Cubic Functiondegree = 3
2 3y x 312
2y x
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Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsQuartic Function
degree = 4
Quintic Functiondegree = 5
4 33.5 5 1y x x x 5 35 5 1y x x x
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End Behavior of End Behavior of EVEN EVEN DEGREE DEGREE
Leading Coefficient Positive
Leading Coefficient Negative
( )
as
f x
x
( )
as
f x
x
( )
as
f x
x
( )
as
f x
x
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End Behavior of End Behavior of ODD ODD DEGREE DEGREE
Leading Coefficient Positive
Leading Coefficient Negative
( )
as
f x
x
( )
as
f x
x
( )
as
f x
x
( )
as
f x
x
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Zeros (or Roots)Zeros (or Roots)The number of REAL ZEROS a polynomial can be determined by how many times the graph crosses the x-axis.
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For each graph, For each graph, - End Behavior?- End Behavior?- Even or Odd?- Even or Odd?
- # of - # of Real Real Zeros? Zeros?
Odd3 Real Zeros
Even1 Real Zeros
Odd3 Real Zeros