Students should be able to… › Evaluate a polynomial function. › Graph a polynomial function.
2.2 Polynomial Function Notes
9
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 Warm-Up • Sketch the graphs of the following: f(x)=x f(x)=x 2 f(x)=x 3 f(x)=x 4 f(x)=x 5 End Behavior: Even functions either start up and end up or start down and end down Odd functions either start down and end up or start up and end down.
-
Upload
lgemgnani -
Category
Technology
-
view
640 -
download
0
description
Transcript of 2.2 Polynomial Function Notes
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1
Warm-Up• Sketch the graphs of the following:
f(x)=x f(x)=x2 f(x)=x3 f(x)=x4 f(x)=x5
End Behavior:Even functions either start up and end up or start down and end down
Odd functions either start down and end up or start up and end down.
Match the equations with their graph.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
21( ) 3 5
2 f x x x
5 4 3( ) 2 2 5 2 f x x x x x
3 2( ) 3 6 f x x x x
4 2( ) 3 5 f x x x
2( ) 4 f x x x
( ) 5 f x x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
A polynomial function is a function of the form1
1 1 0( ) , 0n nn n nf x a x a x a x a a
where n is a nonnegative integer and each ai is a real number.
The polynomial function has a leading coefficient an and degree n.
Examples:5 3( ) 2 3 5 1f x x x x
3 2( ) 6 7f x x x x ( ) 14f x
Section 2.2
Solve the following
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
20 3 2 x x
20 3 2
0 ( 1)( 2)
1 0 2 0
1 2
x x
x x
x x
x x
There are multiple ways to write the answers.
x=1 is a zero
x=1 is a solution
x-1 is a factor
(1,0) is an x-intercept
The correct ways depends on the question.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
A real number a is a zero of a function y = f (x)if and only if f (a) = 0.
A polynomial function of degree n has at most n zeros.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.
1. x = a is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
y
x–2
2
Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2
The real zeros are x = -1,x=0 double root, and x = 2.
When the roots are real the zeros correspond to the x-intercepts. f (x) = x4 – x3 – 2x2
(–1, 0) (0, 0)
(2, 0)
= x2(x2 – x – 2)
= x2(x + 1)(x – 2)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2.
Calc Menu:
The zeros of f(x) are x = – 2, x = 0.5, and x = 1.
– 10 10
10
– 10
Solve for the zeros using a graphing calculator.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
3 21. 3 4 15 20 y x x x
5 22. 13 5 y x x
3 23. 3 8 y x x x
3 24. 8 12 y x x x
Write the polynomial with the following roots.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
1. 3, 2 x
2. 3,0x
3. 2 5, 4 x
4. 3 , 2,0 x double root
