Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook
Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook Analyze Graphs of Polynomial Functions
August 30, 2012
Analyze Graphs of Polynomial Functions
Learning Objecves for CH 2.8 Analyze Graphs of Polynomial Funcons • Zeros • Factors • Soluons • X‐intercept
10 is a zero of the function.
• Local min/max • Turning Points Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
is a factor of the function.
We can prove this with long division.
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Also, the point ( 10 , 0 ) is an xintercept of the function and 10 is a solution of f(x) = 0
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Statements that say a certain value, k, is a : • zero • factor • solution • xintercept are equivalent statements.
Jul 184:17 PM
Jul 184:17 PM
1
Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook Analyze Graphs of Polynomial Functions
In general terms, for the polynomial function
August 30, 2012
Analyze Graphs of Polynomial Functions
Graph the following function:
the following statements are all equivalent: Zero: k is a zero of the polynomial function Factor: xk is a factor of the polynomial function Solution: k is a solution of the polynomial function f(x)=0
Making a table of values will be helpful.
xintercept: If k is a real number, k is an intercept of the graph of the polynomial function f. The graph of f passes through ( k, 0)
Jul 184:17 PM
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
x
4
2
1
0
1
2
4
f(x)
0
4
27/8
2
5/8
0
4
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions local maximum
The graph of possesses special points called the local maximum and local minimum. It is easiest to see these points on the graph and the we can learn how they are defined.
Jul 184:17 PM
local minimum
Jul 184:17 PM
2
Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook Analyze Graphs of Polynomial Functions
Turning Points of a Polynomial Graph
1. The graph of a polynomial function of degree n has at most n1 turning points. 2. If a polynomial function has n distinct real zeros, then its graph has exactly n1 turning points.
Analyze Graphs of Polynomial Functions The graph of a polynomial function of degree n has at most n1 turning points.
Degree = 4 Turning Points = 3
Jul 184:17 PM
Analyze Graphs of Polynomial Functions If a polynomial function has n distinct real zeros, then its graph has exactly n1 turning points.
August 30, 2012
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Learning Objecves for CH 2.8 Analyze Graphs of Polynomial Funcons • Zeros
Zeros = 3 Turning Points = 2
• Factors • Soluons • X‐intercept • Local min/max • Turning Points
Jul 184:17 PM
Jul 184:17 PM
3
August 30, 2012
Analyze Graphs of Polynomial Functions
Learning Objecves for CH 2.8 Analyze Graphs of Polynomial Funcons • Zeros • Factors • Soluons • X‐intercept
10 is a zero of the function.
• Local min/max • Turning Points Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
is a factor of the function.
We can prove this with long division.
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Also, the point ( 10 , 0 ) is an xintercept of the function and 10 is a solution of f(x) = 0
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Statements that say a certain value, k, is a : • zero • factor • solution • xintercept are equivalent statements.
Jul 184:17 PM
Jul 184:17 PM
1
Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook Analyze Graphs of Polynomial Functions
In general terms, for the polynomial function
August 30, 2012
Analyze Graphs of Polynomial Functions
Graph the following function:
the following statements are all equivalent: Zero: k is a zero of the polynomial function Factor: xk is a factor of the polynomial function Solution: k is a solution of the polynomial function f(x)=0
Making a table of values will be helpful.
xintercept: If k is a real number, k is an intercept of the graph of the polynomial function f. The graph of f passes through ( k, 0)
Jul 184:17 PM
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
x
4
2
1
0
1
2
4
f(x)
0
4
27/8
2
5/8
0
4
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Jul 184:17 PM
Analyze Graphs of Polynomial Functions local maximum
The graph of possesses special points called the local maximum and local minimum. It is easiest to see these points on the graph and the we can learn how they are defined.
Jul 184:17 PM
local minimum
Jul 184:17 PM
2
Algebra 2 Ch 2.8 Analyze Graphs of Polynomial Function.notebook Analyze Graphs of Polynomial Functions
Turning Points of a Polynomial Graph
1. The graph of a polynomial function of degree n has at most n1 turning points. 2. If a polynomial function has n distinct real zeros, then its graph has exactly n1 turning points.
Analyze Graphs of Polynomial Functions The graph of a polynomial function of degree n has at most n1 turning points.
Degree = 4 Turning Points = 3
Jul 184:17 PM
Analyze Graphs of Polynomial Functions If a polynomial function has n distinct real zeros, then its graph has exactly n1 turning points.
August 30, 2012
Jul 184:17 PM
Analyze Graphs of Polynomial Functions
Learning Objecves for CH 2.8 Analyze Graphs of Polynomial Funcons • Zeros
Zeros = 3 Turning Points = 2
• Factors • Soluons • X‐intercept • Local min/max • Turning Points
Jul 184:17 PM
Jul 184:17 PM
3
